{"id":42456,"date":"2026-10-13T11:17:48","date_gmt":"2026-10-13T11:17:48","guid":{"rendered":"https:\/\/mp.moonpreneur.com\/math-corner\/?p=42456"},"modified":"2026-10-13T11:23:31","modified_gmt":"2026-10-13T11:23:31","slug":"unit-circle-tangent-values","status":"publish","type":"post","link":"https:\/\/mp.moonpreneur.com\/math-corner\/unit-circle-tangent-values\/","title":{"rendered":"How to Find Tangent Values on the Unit Circle"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"42456\" class=\"elementor elementor-42456\" data-elementor-post-type=\"post\">\n\t\t\t\t\t\t<div class=\"elementor-inner\">\n\t\t\t\t<div class=\"elementor-section-wrap\">\n\t\t\t\t\t\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-4ce9f5b elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"4ce9f5b\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-0df17b4\" data-id=\"0df17b4\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-7655250 elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"7655250\" data-element_type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInDown&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<h2><b>The Circle That Quietly Runs Trigonometry<\/b><\/h2>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-cc7fb5f elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"cc7fb5f\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-9d13dad\" data-id=\"9d13dad\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-d059df2 elementor-widget elementor-widget-text-editor\" data-id=\"d059df2\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<p><span style=\"color: #000000;\"><span style=\"font-weight: 400;\">If trigonometry had a home base, it would be a circle with a radius of exactly 1. Sine and cosine usually get all the spotlight, but tangent is the clever one in the family, the value that tells you how <\/span><i><span style=\"font-weight: 400;\">steep<\/span><\/i><span style=\"font-weight: 400;\"> an angle really is. Once you understand the <\/span>unit circle with tan<span style=\"font-weight: 400;\">, problems in calculus, physics, engineering, and even game design start to feel far less intimidating.<\/span><\/span><\/p><p><span style=\"color: #000000;\"><span style=\"font-weight: 400;\">In this guide, you will learn how to compute tangent values on the unit circle from scratch, how to build a <\/span>full unit circle with tan<span style=\"font-weight: 400;\"> in your own notebook, and how to read a <\/span>unit circle chart with tan<span style=\"font-weight: 400;\"> like a pro. By the end, you will not need to memorize anything. You will simply <\/span><i><span style=\"font-weight: 400;\">see<\/span><\/i><span style=\"font-weight: 400;\"> it.<\/span><\/span><\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-cb3c4c5 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"cb3c4c5\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-60a6246\" data-id=\"60a6246\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-de277a3 elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"de277a3\" data-element_type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInDown&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<h2><b>What Is the Unit Circle?<\/b><\/h2>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-78de8a5 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"78de8a5\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-140e77d\" data-id=\"140e77d\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-8ae4617 elementor-widget elementor-widget-text-editor\" data-id=\"8ae4617\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<p><span style=\"font-weight: 400; color: #000000;\">The unit circle is a circle centered at the origin (0, 0) of a coordinate plane with a radius of 1. Its equation is:<\/span><\/p><p><span style=\"color: #000000;\"><b>x\u00b2 + y\u00b2 = 1<\/b><\/span><\/p><p><span style=\"color: #000000;\"><span style=\"font-weight: 400;\">Every point on this circle can be written as <\/span><b>(cos \u03b8, sin \u03b8)<\/b><span style=\"font-weight: 400;\">, where \u03b8 (theta) is the angle measured counterclockwise from the positive x-axis. That single idea is the foundation of everything that follows:<\/span><\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"color: #000000;\"><span style=\"font-weight: 400;\">The <\/span><b>x-coordinate<\/b><span style=\"font-weight: 400;\"> is the cosine of the angle.<\/span><\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"color: #000000;\"><span style=\"font-weight: 400;\">The <\/span><b>y-coordinate<\/b><span style=\"font-weight: 400;\"> is the sine of the angle.<\/span><\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"color: #000000;\"><span style=\"font-weight: 400;\">The <\/span><b>tangent<\/b><span style=\"font-weight: 400;\"> is built from these two.<\/span><\/span><\/li><\/ul>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-c74c65a elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"c74c65a\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-1432c4c\" data-id=\"1432c4c\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-45f13fa elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"45f13fa\" data-element_type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInDown&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<h2><b>The Golden Formula for Tangent<\/b><\/h2>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-37e34d6 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"37e34d6\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-df33b35\" data-id=\"df33b35\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-6382a28 elementor-widget elementor-widget-text-editor\" data-id=\"6382a28\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<p><span style=\"color: #000000;\"><span style=\"font-weight: 400;\">Here is the one formula that unlocks the entire <\/span><b>unit circle with tan values<\/b><span style=\"font-weight: 400;\">:<\/span><\/span><\/p><p><span style=\"color: #000000;\"><b>tan \u03b8 = sin \u03b8 \u00f7 cos \u03b8 = y \u00f7 x<\/b><\/span><\/p><p><span style=\"font-weight: 400; color: #000000;\">That is it. Tangent is simply the y-coordinate divided by the x-coordinate of the point on the unit circle.<\/span><\/p><h4><span style=\"color: #800080;\"><b>A Geometric Way to Picture It<\/b><\/span><\/h4><p><span style=\"color: #000000;\"><span style=\"font-weight: 400;\">Imagine drawing a vertical line that touches the unit circle at the point (1, 0). Now extend the angle&#8217;s terminal side (the line from the origin through the point on the circle) until it hits that vertical line. The height at which it lands is <\/span><b>tan \u03b8<\/b><span style=\"font-weight: 400;\">. This is where the name comes from: the vertical line is <\/span><i><span style=\"font-weight: 400;\">tangent<\/span><\/i><span style=\"font-weight: 400;\"> to the circle.<\/span><\/span><\/p><p><span style=\"font-weight: 400; color: #000000;\">At small angles the line hits low. As the angle approaches 90\u00b0, the line shoots upward toward infinity, which is why tangent blows up there.<\/span><\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-9c28d34 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"9c28d34\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-8120f7e\" data-id=\"8120f7e\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-de8e56a elementor-widget elementor-widget-html\" data-id=\"de8e56a\" data-element_type=\"widget\" data-widget_type=\"html.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\r\n<div class=\"tangent-guide\">\r\n  <style>\r\n    .tangent-guide {\r\n      font-family: Arial, sans-serif;\r\n      color: #25304a;\r\n      background: linear-gradient(135deg, #f4f2ff, #eef8ff);\r\n      padding: 26px;\r\n      border-radius: 18px;\r\n      line-height: 1.7;\r\n      margin: 20px 0;\r\n    }\r\n    .tangent-guide * { box-sizing: border-box; }\r\n    .tangent-guide h2 {\r\n      color: #5638b5;\r\n      font-size: 25px;\r\n      margin: 0 0 22px;\r\n    }\r\n    .tangent-guide .step {\r\n      background: #fff;\r\n      border: 1px solid #e2def7;\r\n      border-radius: 13px;\r\n      padding: 17px;\r\n      margin: 15px 0;\r\n      box-shadow: 0 4px 12px #3930800b;\r\n    }\r\n    .tangent-guide .step h3 {\r\n      color: #573bb4;\r\n      font-size: 17px;\r\n      margin: 0 0 7px;\r\n    }\r\n    .tangent-guide .formula {\r\n      background: #eeeaff;\r\n      color: #432d99;\r\n      border-left: 4px solid #7455db;\r\n      border-radius: 8px;\r\n      padding: 12px 15px;\r\n      font-size: 18px;\r\n      font-weight: bold;\r\n      text-align: center;\r\n      margin-top: 12px;\r\n    }\r\n    .tangent-guide .table-wrap {\r\n      overflow-x: auto;\r\n      margin-top: 13px;\r\n    }\r\n    .tangent-guide table {\r\n      border-collapse: collapse;\r\n      width: 100%;\r\n      min-width: 390px;\r\n      background: #fff;\r\n      border-radius: 10px;\r\n      overflow: hidden;\r\n      font-size: 14px;\r\n    }\r\n    .tangent-guide th {\r\n      background: #5b43b8;\r\n      color: #fff;\r\n      padding: 12px 9px;\r\n      text-align: left;\r\n    }\r\n    .tangent-guide td {\r\n      padding: 11px 9px;\r\n      border-bottom: 1px solid #e8e7f2;\r\n    }\r\n    .tangent-guide tbody tr:nth-child(even) {\r\n      background: #f5f3ff;\r\n    }\r\n    .tangent-guide .positive {\r\n      color: #16804a;\r\n      font-weight: bold;\r\n    }\r\n    .tangent-guide .negative {\r\n      color: #d34b55;\r\n      font-weight: bold;\r\n    }\r\n    .tangent-guide .memory {\r\n      background: #fff4d8;\r\n      border: 1px solid #f4d68a;\r\n      padding: 15px;\r\n      border-radius: 12px;\r\n      margin-top: 15px;\r\n    }\r\n    .tangent-guide .memory strong { color: #8b5b0a; }\r\n    @media(max-width:480px) {\r\n      .tangent-guide { padding: 15px; }\r\n      .tangent-guide h2 { font-size: 21px; }\r\n    }\r\n  <\/style>\r\n\r\n  <h2>\ud83e\udded Step-by-Step: How to Compute Tangent on the Unit Circle<\/h2>\r\n\r\n  <div class=\"step\">\r\n    <h3>01. Locate the angle<\/h3>\r\n    Start at the positive x-axis and rotate to your angle. Positive angles move counterclockwise, while negative angles move clockwise.\r\n    <div class=\"formula\">Find the angle \u03b8<\/div>\r\n  <\/div>\r\n\r\n  <div class=\"step\">\r\n    <h3>02. Find the coordinates (cos \u03b8, sin \u03b8)<\/h3>\r\n    Use the unit circle to identify the point at your chosen angle.\r\n    <div class=\"formula\">(x, y) = (cos \u03b8, sin \u03b8)<\/div>\r\n  <\/div>\r\n\r\n  <div class=\"step\">\r\n    <h3>03. Divide y by x<\/h3>\r\n    Tangent is the ratio of the y-coordinate to the x-coordinate.\r\n    <div class=\"formula\">tan \u03b8 = sin \u03b8 \u00f7 cos \u03b8 = y \u00f7 x<\/div>\r\n    Simplify the fraction and rationalize the denominator if needed.\r\n  <\/div>\r\n\r\n  <div class=\"step\">\r\n    <h3>04. Check the sign using the quadrant<\/h3>\r\n    The quadrant tells you whether the tangent value is positive or negative.\r\n\r\n    <div class=\"table-wrap\">\r\n      <table>\r\n        <thead>\r\n          <tr>\r\n            <th>Quadrant<\/th>\r\n            <th>Angle Range<\/th>\r\n            <th>sin<\/th>\r\n            <th>cos<\/th>\r\n            <th>tan<\/th>\r\n          <\/tr>\r\n        <\/thead>\r\n        <tbody>\r\n          <tr>\r\n            <td><b>I<\/b><\/td>\r\n            <td>0\u00b0\u201390\u00b0<\/td>\r\n            <td class=\"positive\">+<\/td>\r\n            <td class=\"positive\">+<\/td>\r\n            <td class=\"positive\">+<\/td>\r\n          <\/tr>\r\n          <tr>\r\n            <td><b>II<\/b><\/td>\r\n            <td>90\u00b0\u2013180\u00b0<\/td>\r\n            <td class=\"positive\">+<\/td>\r\n            <td class=\"negative\">\u2212<\/td>\r\n            <td class=\"negative\">\u2212<\/td>\r\n          <\/tr>\r\n          <tr>\r\n            <td><b>III<\/b><\/td>\r\n            <td>180\u00b0\u2013270\u00b0<\/td>\r\n            <td class=\"negative\">\u2212<\/td>\r\n            <td class=\"negative\">\u2212<\/td>\r\n            <td class=\"positive\">+<\/td>\r\n          <\/tr>\r\n          <tr>\r\n            <td><b>IV<\/b><\/td>\r\n            <td>270\u00b0\u2013360\u00b0<\/td>\r\n            <td class=\"negative\">\u2212<\/td>\r\n            <td class=\"positive\">+<\/td>\r\n            <td class=\"negative\">\u2212<\/td>\r\n          <\/tr>\r\n        <\/tbody>\r\n      <\/table>\r\n    <\/div>\r\n\r\n    <div class=\"memory\">\r\n      <strong>\ud83d\udca1 Memory trick: \u201cAll Students Take Calculus\u201d<\/strong><br \/>\r\n      Quadrant I: All are positive. Quadrant II: Only sine is positive. Quadrant III: Only tangent is positive. Quadrant IV: Only cosine is positive.\r\n    <\/div>\r\n  <\/div>\r\n<\/div>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-f2eb3e9 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"f2eb3e9\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-25dd034\" data-id=\"25dd034\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-7ee9689 elementor-widget elementor-widget-html\" data-id=\"7ee9689\" data-element_type=\"widget\" data-widget_type=\"html.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\r\n<div class=\"tangent-examples\">\r\n  <style>\r\n    .tangent-examples {\r\n      font-family: Arial, sans-serif;\r\n      color: #27324a;\r\n      background: #f5f8ff;\r\n      border-radius: 18px;\r\n      padding: 25px;\r\n      margin: 20px 0;\r\n      line-height: 1.7;\r\n    }\r\n    .tangent-examples * { box-sizing: border-box; }\r\n    .tangent-examples h2 {\r\n      font-size: 25px;\r\n      color: #3156b8;\r\n      margin: 0 0 8px;\r\n    }\r\n    .tangent-examples .intro {\r\n      color: #5c6680;\r\n      margin-bottom: 22px;\r\n    }\r\n    .tangent-examples .example {\r\n      background: #fff;\r\n      border: 1px solid #e0e7fb;\r\n      border-radius: 14px;\r\n      padding: 17px;\r\n      margin: 14px 0;\r\n      box-shadow: 0 5px 15px #294c8b0b;\r\n    }\r\n    .tangent-examples .example h3 {\r\n      color: #3156b8;\r\n      font-size: 17px;\r\n      margin: 0 0 8px;\r\n    }\r\n    .tangent-examples .tag {\r\n      display: inline-block;\r\n      font-size: 12px;\r\n      font-weight: bold;\r\n      background: #e8efff;\r\n      color: #3156b8;\r\n      border-radius: 20px;\r\n      padding: 4px 10px;\r\n      margin-bottom: 8px;\r\n    }\r\n    .tangent-examples .equation {\r\n      background: #edf2ff;\r\n      border-radius: 9px;\r\n      padding: 12px;\r\n      margin-top: 11px;\r\n      color: #263e91;\r\n      text-align: center;\r\n      font-weight: bold;\r\n      overflow-wrap: anywhere;\r\n    }\r\n    .tangent-examples .answer {\r\n      background: #e2f8eb;\r\n      color: #146c40;\r\n      border: 1px solid #b9ebce;\r\n      border-radius: 9px;\r\n      padding: 10px;\r\n      text-align: center;\r\n      margin-top: 10px;\r\n      font-weight: bold;\r\n    }\r\n    .tangent-examples .warning {\r\n      background: #fff1e7;\r\n      color: #a44819;\r\n      border: 1px solid #ffd7bd;\r\n      border-radius: 9px;\r\n      padding: 12px;\r\n      margin-top: 10px;\r\n    }\r\n    .tangent-examples .takeaway {\r\n      background: linear-gradient(120deg, #3156b8, #7650c9);\r\n      color: white;\r\n      padding: 18px;\r\n      border-radius: 13px;\r\n      margin-top: 20px;\r\n    }\r\n    .tangent-examples .takeaway h3 {\r\n      margin: 0 0 7px;\r\n      color: white;\r\n    }\r\n    @media(max-width:480px) {\r\n      .tangent-examples { padding: 15px; }\r\n      .tangent-examples h2 { font-size: 21px; }\r\n    }\r\n  <\/style>\r\n\r\n  <h2>\ud83e\uddee Worked Examples: Tangent Values<\/h2>\r\n  <p class=\"intro\">Explore five common angles and see how the unit circle makes each calculation easier.<\/p>\r\n\r\n  <div class=\"example\">\r\n    <span class=\"tag\">EXAMPLE 01<\/span>\r\n    <h3>Find tan 30\u00b0<\/h3>\r\n    The unit-circle coordinates are (\u221a3\/2, 1\/2).\r\n    <div class=\"equation\">\r\n      tan 30\u00b0 = (1\/2) \u00f7 (\u221a3\/2)<br \/>\r\n      = 1\/\u221a3 = \u221a3\/3\r\n    <\/div>\r\n    <div class=\"answer\">Answer: \u221a3\/3<\/div>\r\n  <\/div>\r\n\r\n  <div class=\"example\">\r\n    <span class=\"tag\">EXAMPLE 02<\/span>\r\n    <h3>Find tan 45\u00b0<\/h3>\r\n    The coordinates are (\u221a2\/2, \u221a2\/2). Since x and y are equal, their ratio is 1.\r\n    <div class=\"equation\">\r\n      tan 45\u00b0 = (\u221a2\/2) \u00f7 (\u221a2\/2) = 1\r\n    <\/div>\r\n    <div class=\"answer\">Answer: 1<\/div>\r\n  <\/div>\r\n\r\n  <div class=\"example\">\r\n    <span class=\"tag\">EXAMPLE 03<\/span>\r\n    <h3>Find tan 60\u00b0<\/h3>\r\n    The coordinates are (1\/2, \u221a3\/2).\r\n    <div class=\"equation\">\r\n      tan 60\u00b0 = (\u221a3\/2) \u00f7 (1\/2) = \u221a3\r\n    <\/div>\r\n    <div class=\"answer\">Answer: \u221a3<\/div>\r\n  <\/div>\r\n\r\n  <div class=\"example\">\r\n    <span class=\"tag\">EXAMPLE 04<\/span>\r\n    <h3>Find tan 90\u00b0<\/h3>\r\n    The unit-circle coordinates are (0, 1). The x-coordinate is zero, so the ratio would require division by zero.\r\n    <div class=\"equation\">tan 90\u00b0 = 1 \u00f7 0<\/div>\r\n    <div class=\"warning\">\r\n      <b>Remember:<\/b> tan 90\u00b0 is undefined. The same is true at 270\u00b0, where the tangent graph has a vertical asymptote.\r\n    <\/div>\r\n  <\/div>\r\n\r\n  <div class=\"example\">\r\n    <span class=\"tag\">EXAMPLE 05 \u00b7 QUADRANT II<\/span>\r\n    <h3>Find tan 150\u00b0<\/h3>\r\n    The coordinates are (\u2212\u221a3\/2, 1\/2). The x-coordinate is negative and the y-coordinate is positive.\r\n    <div class=\"equation\">\r\n      tan 150\u00b0 = (1\/2) \u00f7 (\u2212\u221a3\/2)<br \/>\r\n      = \u22121\/\u221a3 = \u2212\u221a3\/3\r\n    <\/div>\r\n    <div class=\"answer\">Answer: \u2212\u221a3\/3<\/div>\r\n  <\/div>\r\n\r\n  <div class=\"takeaway\">\r\n    <h3>\u2b50 Quick takeaway<\/h3>\r\n    To find tangent on the unit circle, divide the y-coordinate by the x-coordinate. If the x-coordinate is zero, tangent is undefined. Always check the quadrant to confirm the sign.\r\n  <\/div>\r\n<\/div>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-6c58119 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"6c58119\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-94fa38a\" data-id=\"94fa38a\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-e95f753 elementor-widget elementor-widget-html\" data-id=\"e95f753\" data-element_type=\"widget\" data-widget_type=\"html.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\r\n<div class=\"tan-chart\">\r\n<style>\r\n.tan-chart{font-family:Arial,sans-serif;color:#263149;background:#f5f7ff;padding:24px;border-radius:18px;line-height:1.6;margin:20px 0}\r\n.tan-chart *{box-sizing:border-box}\r\n.tan-chart h2{color:#4934a3;font-size:25px;margin:0 0 10px}\r\n.tan-chart p{margin:8px 0 16px}\r\n.tan-chart .chart-note{background:#eae7ff;padding:13px 15px;border-left:4px solid #7054d6;border-radius:8px;margin:15px 0}\r\n.tan-chart h3{font-size:17px;color:#4934a3;margin:24px 0 10px}\r\n.tan-chart .table-wrap{overflow-x:auto;border-radius:10px;border:1px solid #dcdff0;background:#fff}\r\n.tan-chart table{width:100%;min-width:480px;border-collapse:collapse;font-size:14px}\r\n.tan-chart th{background:#4e3ba4;color:white;text-align:left;padding:12px 10px}\r\n.tan-chart td{padding:10px;border-bottom:1px solid #e7e9f3}\r\n.tan-chart tbody tr:nth-child(even){background:#f0efff}\r\n.tan-chart tbody tr:hover{background:#e6f2ff}\r\n.tan-chart .value{font-weight:bold;color:#246d51}\r\n.tan-chart .undefined{color:#c14b3c;font-weight:bold}\r\n.tan-chart .section-label{display:inline-block;background:#ddd7ff;color:#4833a0;padding:5px 10px;border-radius:20px;font-size:12px;font-weight:bold;margin-bottom:8px}\r\n.tan-chart .footnote{font-size:13px;color:#626b80;margin-top:12px}\r\n@media(max-width:500px){.tan-chart{padding:15px}.tan-chart h2{font-size:21px}}\r\n<\/style>\r\n\r\n<h2>\ud83d\udcd0 The Full Unit Circle With Tan: Complete Chart<\/h2>\r\n<p>Explore tangent values for special angles from 0\u00b0 to 360\u00b0. Use this chart to spot patterns, review coordinates, and build confidence with trigonometry.<\/p>\r\n\r\n<div class=\"chart-note\">\r\n<strong>Key formula:<\/strong> tan \u03b8 = sin \u03b8 \/ cos \u03b8 = y \/ x\r\n<br \/>Remember: tangent is undefined whenever cos \u03b8 = 0.\r\n<\/div>\r\n\r\n<span class=\"section-label\">QUADRANTS I &amp; II<\/span>\r\n<h3>Unit Circle Chart With Tan<\/h3>\r\n<div class=\"table-wrap\">\r\n<table>\r\n<thead><tr><th>Degrees<\/th><th>Radians<\/th><th>Coordinates (cos \u03b8, sin \u03b8)<\/th><th>tan \u03b8<\/th><\/tr><\/thead>\r\n<tbody>\r\n<tr><td>0\u00b0<\/td><td>0<\/td><td>(1, 0)<\/td><td class=\"value\">0<\/td><\/tr>\r\n<tr><td>30\u00b0<\/td><td>\u03c0\/6<\/td><td>(\u221a3\/2, 1\/2)<\/td><td class=\"value\">\u221a3\/3<\/td><\/tr>\r\n<tr><td>45\u00b0<\/td><td>\u03c0\/4<\/td><td>(\u221a2\/2, \u221a2\/2)<\/td><td class=\"value\">1<\/td><\/tr>\r\n<tr><td>60\u00b0<\/td><td>\u03c0\/3<\/td><td>(1\/2, \u221a3\/2)<\/td><td class=\"value\">\u221a3<\/td><\/tr>\r\n<tr><td>90\u00b0<\/td><td>\u03c0\/2<\/td><td>(0, 1)<\/td><td class=\"undefined\">Undefined<\/td><\/tr>\r\n<tr><td>120\u00b0<\/td><td>2\u03c0\/3<\/td><td>(\u22121\/2, \u221a3\/2)<\/td><td class=\"value\">\u2212\u221a3<\/td><\/tr>\r\n<tr><td>135\u00b0<\/td><td>3\u03c0\/4<\/td><td>(\u2212\u221a2\/2, \u221a2\/2)<\/td><td class=\"value\">\u22121<\/td><\/tr>\r\n<tr><td>150\u00b0<\/td><td>5\u03c0\/6<\/td><td>(\u2212\u221a3\/2, 1\/2)<\/td><td class=\"value\">\u2212\u221a3\/3<\/td><\/tr>\r\n<tr><td>180\u00b0<\/td><td>\u03c0<\/td><td>(\u22121, 0)<\/td><td class=\"value\">0<\/td><\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n\r\n<span class=\"section-label\" style=\"margin-top:24px\">QUADRANTS III &amp; IV<\/span>\r\n<h3>Continue the Tangent Chart<\/h3>\r\n<div class=\"table-wrap\">\r\n<table>\r\n<thead><tr><th>Degrees<\/th><th>Radians<\/th><th>Coordinates (cos \u03b8, sin \u03b8)<\/th><th>tan \u03b8<\/th><\/tr><\/thead>\r\n<tbody>\r\n<tr><td>210\u00b0<\/td><td>7\u03c0\/6<\/td><td>(\u2212\u221a3\/2, \u22121\/2)<\/td><td class=\"value\">\u221a3\/3<\/td><\/tr>\r\n<tr><td>225\u00b0<\/td><td>5\u03c0\/4<\/td><td>(\u2212\u221a2\/2, \u2212\u221a2\/2)<\/td><td class=\"value\">1<\/td><\/tr>\r\n<tr><td>240\u00b0<\/td><td>4\u03c0\/3<\/td><td>(\u22121\/2, \u2212\u221a3\/2)<\/td><td class=\"value\">\u221a3<\/td><\/tr>\r\n<tr><td>270\u00b0<\/td><td>3\u03c0\/2<\/td><td>(0, \u22121)<\/td><td class=\"undefined\">Undefined<\/td><\/tr>\r\n<tr><td>300\u00b0<\/td><td>5\u03c0\/3<\/td><td>(1\/2, \u2212\u221a3\/2)<\/td><td class=\"value\">\u2212\u221a3<\/td><\/tr>\r\n<tr><td>315\u00b0<\/td><td>7\u03c0\/4<\/td><td>(\u221a2\/2, \u2212\u221a2\/2)<\/td><td class=\"value\">\u22121<\/td><\/tr>\r\n<tr><td>330\u00b0<\/td><td>11\u03c0\/6<\/td><td>(\u221a3\/2, \u22121\/2)<\/td><td class=\"value\">\u2212\u221a3\/3<\/td><\/tr>\r\n<tr><td>360\u00b0<\/td><td>2\u03c0<\/td><td>(1, 0)<\/td><td class=\"value\">0<\/td><\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n\r\n<p class=\"footnote\">Study tip: Learn the values at 30\u00b0, 45\u00b0, and 60\u00b0 first. Then use quadrant signs to determine the values of related angles.<\/p>\r\n<\/div>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-04bea53 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"04bea53\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-1ca154b\" data-id=\"1ca154b\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-1927e71 elementor-widget elementor-widget-html\" data-id=\"1927e71\" data-element_type=\"widget\" data-widget_type=\"html.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\r\n<div class=\"tan-patterns\">\r\n<style>\r\n.tan-patterns{font-family:Arial,sans-serif;color:#27324a;background:linear-gradient(145deg,#f2f5ff,#f9f3ff);padding:25px;border-radius:18px;line-height:1.7;margin:20px 0}\r\n.tan-patterns *{box-sizing:border-box}\r\n.tan-patterns h2{font-size:25px;color:#4d37a4;margin:0 0 10px}\r\n.tan-patterns .intro{color:#59647b;margin-bottom:22px}\r\n.tan-patterns .pattern{background:#fff;border:1px solid #e2e2f4;border-radius:13px;padding:17px;margin:14px 0;box-shadow:0 4px 14px #3730800a}\r\n.tan-patterns .pattern h3{font-size:17px;color:#4d37a4;margin:0 0 8px}\r\n.tan-patterns .number{display:inline-flex;align-items:center;justify-content:center;background:#4d37a4;color:white;width:30px;height:30px;border-radius:9px;margin-right:7px;font-size:14px}\r\n.tan-patterns .math{background:#efedff;color:#44319b;text-align:center;padding:13px;border-radius:9px;margin:12px 0;font-weight:bold;font-size:16px}\r\n.tan-patterns .mini-grid{display:grid;grid-template-columns:repeat(4,minmax(0,1fr));gap:8px;margin:12px 0}\r\n.tan-patterns .mini{background:#f0efff;border-radius:9px;padding:10px 5px;text-align:center;font-size:13px}\r\n.tan-patterns .mini strong{display:block;color:#4936a0;font-size:16px;margin-top:3px}\r\n.tan-patterns .tip{background:#e1f7ec;border-left:4px solid #2b9b69;padding:13px;border-radius:8px;margin-top:12px}\r\n.tan-patterns .summary{background:linear-gradient(120deg,#44339e,#7552ce);color:white;padding:18px;border-radius:13px;margin-top:20px}\r\n.tan-patterns .summary h3{color:white;margin:0 0 8px}\r\n@media(max-width:480px){.tan-patterns{padding:15px}.tan-patterns h2{font-size:21px}.tan-patterns .mini-grid{grid-template-columns:repeat(2,minmax(0,1fr))}}\r\n<\/style>\r\n\r\n<h2>\ud83e\udde0 Patterns That Make Memorizing Unnecessary<\/h2>\r\n<p class=\"intro\">The unit circle is full of repeating patterns. Understand these four rules, and you can find many tangent values without memorizing the entire chart.<\/p>\r\n\r\n<div class=\"pattern\">\r\n<h3><span class=\"number\">1<\/span>Reference Angles Do the Heavy Lifting<\/h3>\r\nEvery angle has a reference angle in Quadrant I. Its tangent has the same magnitude as the reference angle's tangent; only the sign changes based on the quadrant.\r\n<div class=\"math\">tan 30\u00b0 = \u221a3\/3<br \/>tan 210\u00b0 = \u221a3\/3<\/div>\r\nBoth angles have a 30\u00b0 reference angle, and tangent is positive in Quadrant III.\r\n<div class=\"tip\"><strong>Remember:<\/strong> Find the reference angle first, then check the quadrant for the sign.<\/div>\r\n<\/div>\r\n\r\n<div class=\"pattern\">\r\n<h3><span class=\"number\">2<\/span>Tangent Repeats Every 180\u00b0<\/h3>\r\nUnlike sine and cosine, tangent repeats after every 180\u00b0 (\u03c0 radians).\r\n<div class=\"math\">tan(\u03b8 + 180\u00b0) = tan \u03b8<\/div>\r\nFor example:\r\n<div class=\"math\">tan 30\u00b0 = tan 210\u00b0 = \u221a3\/3<\/div>\r\nAdding 180\u00b0 gives the same tangent value wherever both expressions are defined.\r\n<\/div>\r\n\r\n<div class=\"pattern\">\r\n<h3><span class=\"number\">3<\/span>The First-Quadrant Sequence<\/h3>\r\nFor the special angles from 0\u00b0 to 90\u00b0, tangent follows a predictable sequence.\r\n<div class=\"mini-grid\">\r\n<div class=\"mini\">0\u00b0<strong>0<\/strong><\/div>\r\n<div class=\"mini\">30\u00b0<strong>\u221a3\/3<\/strong><\/div>\r\n<div class=\"mini\">45\u00b0<strong>1<\/strong><\/div>\r\n<div class=\"mini\">60\u00b0<strong>\u221a3<\/strong><\/div>\r\n<\/div>\r\n<div class=\"math\">90\u00b0 \u2192 Undefined<\/div>\r\nThe values increase from 0 to \u221a3, then tangent becomes undefined at 90\u00b0.\r\n<\/div>\r\n\r\n<div class=\"pattern\">\r\n<h3><span class=\"number\">4<\/span>Symmetry Through the Origin<\/h3>\r\nTangent is an odd function, meaning that changing an angle to its opposite changes the sign of the tangent.\r\n<div class=\"math\">tan(\u2212\u03b8) = \u2212tan \u03b8<\/div>\r\nFor example:\r\n<div class=\"math\">tan(\u221245\u00b0) = \u2212tan 45\u00b0 = \u22121<\/div>\r\nThis works because sine changes sign for opposite angles while cosine stays the same.\r\n<\/div>\r\n\r\n<div class=\"summary\">\r\n<h3>\u2728 Your Quick Revision<\/h3>\r\n<strong>Reference angle:<\/strong> Find the familiar angle.<br \/>\r\n<strong>Quadrant:<\/strong> Determine the sign.<br \/>\r\n<strong>Period:<\/strong> Add or subtract 180\u00b0 to find equivalent tangent values.<br \/>\r\n<strong>Undefined values:<\/strong> Tangent is undefined at 90\u00b0 and 270\u00b0 within one full rotation.\r\n<\/div>\r\n<\/div>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-a5f496e elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"a5f496e\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-6cd76ff\" data-id=\"6cd76ff\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-0239a13 elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"0239a13\" data-element_type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInDown&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<h2><b>A Quick Memory Trick: The Hand Method<\/b><\/h2>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-9263af7 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"9263af7\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-fa1f28e\" data-id=\"fa1f28e\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-75538a5 elementor-widget elementor-widget-text-editor\" data-id=\"75538a5\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<p><span style=\"font-weight: 400;\">Hold up your left hand with your palm facing you. Assign the fingers from thumb to pinky the angles <\/span><b>0\u00b0, 30\u00b0, 45\u00b0, 60\u00b0, 90\u00b0<\/b><span style=\"font-weight: 400;\">. To find the tangent of an angle, fold down that finger. Then:<\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">sin \u03b8 = \u221a(fingers <\/span><i><span style=\"font-weight: 400;\">below<\/span><\/i><span style=\"font-weight: 400;\"> the folded finger) \u00f7 2<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">cos \u03b8 = \u221a(fingers <\/span><i><span style=\"font-weight: 400;\">above<\/span><\/i><span style=\"font-weight: 400;\"> the folded finger) \u00f7 2<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">tan \u03b8 = sin \u03b8 \u00f7 cos \u03b8<\/span><\/li><\/ul><p><span style=\"font-weight: 400;\">For 60\u00b0, you fold the ring finger. Three fingers sit on one side and one on the other, giving sin = \u221a3\/2 and cos = \u221a1\/2 = 1\/2, so tan 60\u00b0 = \u221a3. It feels like a party trick, but it works every time.<\/span><\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-5a8a557 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"5a8a557\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-182f4ff\" data-id=\"182f4ff\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-2c259b2 elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"2c259b2\" data-element_type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInDown&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<h2><b>Common Mistakes to Avoid<\/b><\/h2>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-975cb08 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"975cb08\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-9ac9d2c\" data-id=\"9ac9d2c\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-c5fbed3 elementor-widget elementor-widget-text-editor\" data-id=\"c5fbed3\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<ol><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"color: #000000;\"><b>Dividing x by y instead of y by x.<\/b><span style=\"font-weight: 400;\"> Tangent is always sin over cos, never the reverse (that is cotangent).<\/span><\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"color: #000000;\"><b>Forgetting the sign.<\/b><span style=\"font-weight: 400;\"> Always check the quadrant before finalizing your answer.<\/span><\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"color: #000000;\"><b>Writing 0 for tan 90\u00b0.<\/b><span style=\"font-weight: 400;\"> It is undefined, not zero.<\/span><\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"color: #000000;\"><b>Mixing up degrees and radians.<\/b><span style=\"font-weight: 400;\"> Confirm which mode your calculator is in.<\/span><\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"color: #000000;\"><b>Not rationalizing.<\/b><span style=\"font-weight: 400;\"> Many teachers prefer \u221a3\/3 over 1\/\u221a3.<\/span><\/span><\/li><\/ol>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-f51f1cc elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"f51f1cc\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-2fcea2e\" data-id=\"2fcea2e\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-6dfe684 elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"6dfe684\" data-element_type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInDown&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<h2><b>Quick Practice Problems<\/b><\/h2>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-e65ed20 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"e65ed20\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-405f684\" data-id=\"405f684\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-4e37ebb elementor-widget elementor-widget-text-editor\" data-id=\"4e37ebb\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<p><span style=\"font-weight: 400; color: #000000;\">Try these before looking at the answers.<\/span><\/p><ol><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400; color: #000000;\">Find tan 120\u00b0.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400; color: #000000;\">Find tan 315\u00b0.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400; color: #000000;\">Find tan 7\u03c0\/6.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400; color: #000000;\">Is tan 270\u00b0 defined?<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400; color: #000000;\">What is tan(\u221260\u00b0)?<\/span><\/li><\/ol><p><span style=\"color: #000000;\"><b>Answers:<\/b><span style=\"font-weight: 400;\"> 1) \u2212\u221a3\u00a0 2) \u22121\u00a0 3) \u221a3\/3\u00a0 4) No, it is undefined\u00a0 5) \u2212\u221a3<\/span><\/span><\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-86cb0af elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"86cb0af\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-da0fb6e\" data-id=\"da0fb6e\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-9d676bc elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"9d676bc\" data-element_type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInDown&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<h3><b>Conclusion<\/b><\/h3>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-defa53e elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"defa53e\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-7b79c80\" data-id=\"7b79c80\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-3934bb2 elementor-widget elementor-widget-text-editor\" data-id=\"3934bb2\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-text-editor elementor-clearfix\">\n\t\t\t\t<p><span style=\"font-weight: 400;\">Computing the <\/span>unit circle with tan values<span style=\"font-weight: 400;\"> comes down to three simple moves: find the coordinates, divide y by x, and check the sign by quadrant. Once you see that tangent is just the slope of the line from the origin to a point on the circle, the entire <\/span>full unit circle with tan<span style=\"font-weight: 400;\"> stops being a wall of numbers and becomes a pattern you can rebuild anytime.<\/span><\/p><p><span style=\"font-weight: 400;\">Keep this <\/span>unit circle chart with tan<span style=\"font-weight: 400;\"> handy, practice with the problems above, and soon you will be flying through trigonometry with confidence.<\/span><\/p><p><i><span style=\"font-weight: 400;\">Happy calculating!<\/span><\/i><\/p><p>You can opt for our <a href=\"https:\/\/moonpreneur.com\/innovator-program\/advanced-math\/\">Advanced Math<\/a>\u00a0or Vedic Math+Mental Math courses. Our\u00a0<a href=\"https:\/\/mp.moonpreneur.com\/math-quiz-for-kids\/\">Math Quiz<\/a> for grades 3rd, 4th, 5th, and 6th helps in further exciting and engaging in mathematics with hands-on lessons.<\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-e964505 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"e964505\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t\t\t<div class=\"elementor-row\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-30ad7bc\" data-id=\"30ad7bc\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-column-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-b5ddfb3 elementor-widget elementor-widget-html\" data-id=\"b5ddfb3\" data-element_type=\"widget\" data-widget_type=\"html.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\r\n<div class=\"unit-circle-faq\">\r\n  <style>\r\n    .unit-circle-faq {\r\n      font-family: Arial, sans-serif;\r\n      background: linear-gradient(135deg, #f5f3ff, #eff8ff);\r\n      color: #29334b;\r\n      padding: 25px;\r\n      border-radius: 18px;\r\n      margin: 20px 0;\r\n      line-height: 1.7;\r\n    }\r\n\r\n    .unit-circle-faq * {\r\n      box-sizing: border-box;\r\n    }\r\n\r\n    .unit-circle-faq .faq-heading {\r\n      display: flex;\r\n      align-items: center;\r\n      gap: 12px;\r\n      margin-bottom: 22px;\r\n    }\r\n\r\n    .unit-circle-faq .faq-icon {\r\n      width: 46px;\r\n      height: 46px;\r\n      flex-shrink: 0;\r\n      display: flex;\r\n      justify-content: center;\r\n      align-items: center;\r\n      border-radius: 13px;\r\n      background: #5b45b5;\r\n      color: white;\r\n      font-size: 25px;\r\n      font-weight: bold;\r\n    }\r\n\r\n    .unit-circle-faq h2 {\r\n      color: #48339c;\r\n      font-size: 24px;\r\n      margin: 0;\r\n    }\r\n\r\n    .unit-circle-faq .faq-item {\r\n      background: #fff;\r\n      border: 1px solid #e5e1f7;\r\n      border-radius: 12px;\r\n      padding: 16px 18px;\r\n      margin-bottom: 13px;\r\n      box-shadow: 0 4px 12px rgba(62, 49, 130, 0.05);\r\n    }\r\n\r\n    .unit-circle-faq .faq-item:nth-of-type(2) {\r\n      border-left: 4px solid #8065dc;\r\n    }\r\n\r\n    .unit-circle-faq .faq-item:nth-of-type(3) {\r\n      border-left: 4px solid #3396ce;\r\n    }\r\n\r\n    .unit-circle-faq .faq-item:nth-of-type(4) {\r\n      border-left: 4px solid #26a579;\r\n    }\r\n\r\n    .unit-circle-faq .faq-item:nth-of-type(5) {\r\n      border-left: 4px solid #e6a33a;\r\n    }\r\n\r\n    .unit-circle-faq h3 {\r\n      color: #343064;\r\n      font-size: 16px;\r\n      line-height: 1.5;\r\n      margin: 0 0 7px;\r\n    }\r\n\r\n    .unit-circle-faq p {\r\n      font-size: 14px;\r\n      margin: 0;\r\n      color: #505b71;\r\n    }\r\n\r\n    .unit-circle-faq .answer-highlight {\r\n      color: #4936a1;\r\n      font-weight: bold;\r\n    }\r\n\r\n    .unit-circle-faq .formula {\r\n      display: inline-block;\r\n      background: #eeebff;\r\n      color: #4b35a1;\r\n      border-radius: 5px;\r\n      padding: 2px 7px;\r\n      font-weight: bold;\r\n      white-space: normal;\r\n    }\r\n\r\n    .unit-circle-faq .bottom-tip {\r\n      background: #e5f7ed;\r\n      color: #246849;\r\n      padding: 13px 15px;\r\n      border-radius: 10px;\r\n      margin-top: 18px;\r\n      font-size: 14px;\r\n    }\r\n\r\n    @media (max-width: 480px) {\r\n      .unit-circle-faq {\r\n        padding: 16px;\r\n      }\r\n\r\n      .unit-circle-faq h2 {\r\n        font-size: 21px;\r\n      }\r\n\r\n      .unit-circle-faq .faq-item {\r\n        padding: 14px;\r\n      }\r\n    }\r\n  <\/style>\r\n\r\n  <div class=\"faq-heading\">\r\n    <div class=\"faq-icon\">?<\/div>\r\n    <h2>Frequently Asked Questions<\/h2>\r\n  <\/div>\r\n\r\n  <div class=\"faq-item\">\r\n    <h3>What is the unit circle with tan?<\/h3>\r\n    <p>The unit circle is a circle with radius 1 centered at the origin. It helps you find tangent values using the sine and cosine of an angle. The key formula is <span class=\"formula\">tan \u03b8 = sin \u03b8 \/ cos \u03b8<\/span>.<\/p>\r\n  <\/div>\r\n\r\n  <div class=\"faq-item\">\r\n    <h3>Why is tan 90\u00b0 undefined?<\/h3>\r\n    <p>At 90\u00b0, cosine equals 0. Since tangent requires dividing sine by cosine, calculating <span class=\"formula\">1 \/ 0<\/span> is impossible. Therefore, tan 90\u00b0 is undefined.<\/p>\r\n  <\/div>\r\n\r\n  <div class=\"faq-item\">\r\n    <h3>How many tangent values do I really need to memorize?<\/h3>\r\n    <p>Start with the first-quadrant values: <span class=\"formula\">0, \u221a3\/3, 1, \u221a3<\/span>, and the fact that tan 90\u00b0 is undefined. You can use reference angles and quadrant signs to find many other values.<\/p>\r\n  <\/div>\r\n\r\n  <div class=\"faq-item\">\r\n    <h3>Is tangent positive or negative in Quadrant III?<\/h3>\r\n    <p><span class=\"answer-highlight\">Tangent is positive in Quadrant III.<\/span> Both sine and cosine are negative, so their ratio is positive: a negative divided by a negative gives a positive result.<\/p>\r\n  <\/div>\r\n\r\n  <div class=\"bottom-tip\">\r\n    <strong>Quick tip:<\/strong> Remember that tangent is positive in Quadrants I and III, negative in Quadrants II and IV, and undefined wherever cosine equals zero.\r\n  <\/div>\r\n<\/div>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>The Circle That Quietly Runs Trigonometry If trigonometry had a home base, it would be a circle with a radius of exactly 1. Sine and cosine usually get all the spotlight, but tangent is the clever one in the family, the value that tells you how steep an angle really is. Once you understand the [&hellip;]<\/p>\n","protected":false},"author":116,"featured_media":42454,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[969],"tags":[],"acf":[],"_links":{"self":[{"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/posts\/42456"}],"collection":[{"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/users\/116"}],"replies":[{"embeddable":true,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/comments?post=42456"}],"version-history":[{"count":4,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/posts\/42456\/revisions"}],"predecessor-version":[{"id":42462,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/posts\/42456\/revisions\/42462"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/media\/42454"}],"wp:attachment":[{"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/media?parent=42456"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/categories?post=42456"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mp.moonpreneur.com\/math-corner\/wp-json\/wp\/v2\/tags?post=42456"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}