{"id":12017,"date":"2026-03-09T14:26:42","date_gmt":"2026-03-09T05:26:42","guid":{"rendered":"https:\/\/www.manabi-edekeru.com\/?p=12017"},"modified":"2026-07-18T11:38:59","modified_gmt":"2026-07-18T02:38:59","slug":"trigonometric-functions","status":"publish","type":"post","link":"https:\/\/www.manabi-edekeru.com\/en\/learning\/mathematics\/trigonometric-functions\/","title":{"rendered":"Complete Guide to Trigonometric Functions (sin, cos, tan)"},"content":{"rendered":"\n<div class=\"wrapper\">\n<div class=\"sidebar\">\n\n\n<nav id=\"TOC\" role=\"doc-toc\">\n    <h2 id=\"toc-title\">On this page<\/h2>\n   \n  <ul>\n  <li><a href=\"#trigonometric-ratios\" id=\"toc-trigonometric-ratios\"><span class=\"header-section-number\">1<\/span> Trigonometric Ratios<\/a><\/li>\n  <li><a href=\"#sec-def-trigonometric-functions\" id=\"toc-sec-def-trigonometric-functions\"><span class=\"header-section-number\">2<\/span> Definition of Trigonometric Functions<\/a><\/li>\n  <li><a href=\"#relationship-between-trigonometric-functions-and-ratios\" id=\"toc-relationship-between-trigonometric-functions-and-ratios\"><span class=\"header-section-number\">3<\/span> Relationship Between Trigonometric Functions and Ratios<\/a><\/li>\n  <li><a href=\"#degrees-and-radians\" id=\"toc-degrees-and-radians\"><span class=\"header-section-number\">4<\/span> Degrees and Radians<\/a><\/li>\n  <li><a href=\"#sec-trig-independent-of-radii\" id=\"toc-sec-trig-independent-of-radii\"><span class=\"header-section-number\">5<\/span> Trigonometric Functions Are Independent of the Radius<\/a><\/li>\n  <li><a href=\"#sec-calclulate-trigonometric-functions\" id=\"toc-sec-calclulate-trigonometric-functions\"><span class=\"header-section-number\">6<\/span> Calculating Trigonometric Functions<\/a><\/li>\n  <li><a href=\"#trigonometric-identities\" id=\"toc-trigonometric-identities\"><span class=\"header-section-number\">7<\/span> Trigonometric Identities<\/a><\/li>\n  <li><a href=\"#graphs-of-trigonometric-functions\" id=\"toc-graphs-of-trigonometric-functions\"><span class=\"header-section-number\">8<\/span> Graphs of Trigonometric Functions<\/a>\n  <ul>\n  <li><a href=\"#periodicity-of-trigonometric-functions\" id=\"toc-periodicity-of-trigonometric-functions\"><span class=\"header-section-number\">8.1<\/span> Periodicity of Trigonometric Functions<\/a><\/li>\n  <li><a href=\"#amplitude-maximum-and-minimum-values-of-trigonometric-functions\" id=\"toc-amplitude-maximum-and-minimum-values-of-trigonometric-functions\"><span class=\"header-section-number\">8.2<\/span> Amplitude, Maximum and Minimum Values of Trigonometric Functions<\/a><\/li>\n  <li><a href=\"#domain-and-range-of-trigonometric-functions\" id=\"toc-domain-and-range-of-trigonometric-functions\"><span class=\"header-section-number\">8.3<\/span> Domain and Range of Trigonometric Functions<\/a><\/li>\n  <li><a href=\"#even-and-odd-functions-in-trigonometry\" id=\"toc-even-and-odd-functions-in-trigonometry\"><span class=\"header-section-number\">8.4<\/span> Even and Odd Functions in Trigonometry<\/a><\/li>\n  <\/ul><\/li>\n  <li><a href=\"#parallel-translation-stretching-and-shrinking-of-trigonometric-graphs\" id=\"toc-parallel-translation-stretching-and-shrinking-of-trigonometric-graphs\"><span class=\"header-section-number\">9<\/span> Parallel Translation, Stretching and Shrinking of Trigonometric Graphs<\/a>\n  <ul>\n  <li><a href=\"#parallel-translation-of-trigonometric-graphs\" id=\"toc-parallel-translation-of-trigonometric-graphs\"><span class=\"header-section-number\">9.1<\/span> Parallel Translation of Trigonometric Graphs<\/a><\/li>\n  <li><a href=\"#scaling-of-trigonometric-graphs\" id=\"toc-scaling-of-trigonometric-graphs\"><span class=\"header-section-number\">9.2<\/span> Scaling of Trigonometric Graphs<\/a><\/li>\n  <li><a href=\"#summary-of-parallel-translation-and-scaling-of-trigonometric-graphs\" id=\"toc-summary-of-parallel-translation-and-scaling-of-trigonometric-graphs\"><span class=\"header-section-number\">9.3<\/span> Summary of Parallel Translation and Scaling of Trigonometric Graphs<\/a><\/li>\n  <\/ul><\/li>\n  <li><a href=\"#conclusion\" id=\"toc-conclusion\"><span class=\"header-section-number\">10<\/span> Conclusion<\/a><\/li>\n  <\/ul>\n<\/div>\n<\/nav>\n<div class=\"main-content\">\n<h2 data-number=\"1\" id=\"trigonometric-ratios\" class=\"anchored\"><span class=\"header-section-number\">1<\/span> Trigonometric Ratios<\/h2>\n<p>Trigonometric functions extend the concept of trigonometric ratios. Trigonometric ratios relate the ratios of the sides of a right-angled triangle to its angles. As explained in <a href=\"#sec-def-trigonometric-functions\" class=\"quarto-xref\">Section&nbsp;2<\/a>, trigonometric functions are defined using the circle; however, the fundamental idea still relies on considering right-angled triangles in trigonometric ratios. Before learning trigonometric functions, it is useful to have a brief understanding of trigonometric ratios.<\/p>\n<p>Trigonometric ratios relate the angles and side ratios of the right-angled triangle shown in <a href=\"#fig-def-trigonometric-ratios\" class=\"quarto-xref\">Figure&nbsp;1<\/a>. Specifically, \u201c<span class=\"math inline\">\\sin \\theta<\/span> represents the ratio of the side opposite to <span class=\"math inline\">\\theta<\/span> to the hypotenuse,\u201d \u201c<span class=\"math inline\">\\cos \\theta<\/span> is the ratio of the side adjacent to <span class=\"math inline\">\\theta<\/span> to the hypotenuse,\u201d and \u201c<span class=\"math inline\">\\tan \\theta<\/span> represents the ratio of the two legs other than the hypotenuse.\u201d<\/p>\n<div id=\"fig-def-trigonometric-ratios\" class=\"quarto-float quarto-figure quarto-figure-center anchored\">\n<figure class=\"quarto-float quarto-float-fig\">\n<div aria-describedby=\"fig-def-trigonometric-ratios-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca\">\n<img decoding=\"async\" src=\"https:\/\/www.manabi-edekeru.com\/wp-content\/uploads\/trigonometric-ratios-fig-def-trigonometric-ratios.webp\">\n<\/div>\n<figcaption class=\"quarto-float-caption-bottom quarto-float-caption quarto-float-fig\" id=\"fig-def-trigonometric-ratios-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca\">\nFigure&nbsp;1: <em>Definition of trigonometric ratios<\/em>\n<\/figcaption>\n<\/figure>\n<\/div>\n<h2 data-number=\"2\" id=\"sec-def-trigonometric-functions\" class=\"anchored\"><span class=\"header-section-number\">2<\/span> Definition of Trigonometric Functions<\/h2>\n<p>Trigonometric functions are defined using the circle as shown in <a href=\"#fig-def-trigonometric-funtions\" class=\"quarto-xref\">Figure&nbsp;2<\/a>.<\/p>\n<div id=\"fig-def-trigonometric-funtions\" class=\"quarto-float quarto-figure quarto-figure-center anchored\">\n<figure class=\"quarto-float quarto-float-fig\">\n<div aria-describedby=\"fig-def-trigonometric-funtions-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca\">\n<img decoding=\"async\" src=\"https:\/\/www.manabi-edekeru.com\/wp-content\/uploads\/fig-class-memo-trig-formlas-trig-def.png\">\n<\/div>\n<figcaption class=\"quarto-float-caption-bottom quarto-float-caption quarto-float-fig\" id=\"fig-def-trigonometric-funtions-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca\">\nFigure&nbsp;2: Definition of trigonometric functions\n<\/figcaption>\n<\/figure>\n<\/div>\n<p>Consider a circle of radius <span class=\"math inline\">r<\/span>. Measure the angle <span class=\"math inline\">\\theta<\/span> counterclockwise from the positive direction of the <span class=\"math inline\">x<\/span>-axis and plot the point <span class=\"math inline\">A(x, y)<\/span> on the circumference. Using the radius <span class=\"math inline\">r<\/span> and the coordinates <span class=\"math inline\">x<\/span> and <span class=\"math inline\">y<\/span> of point <span class=\"math inline\">A<\/span>, trigonometric functions are defined:<\/p>\n<p><span class=\"math display\">\n\\begin{aligned}\n\\sin \\theta &amp;= \\frac{y}{r}\\\\\n\\cos \\theta &amp;= \\frac{x}{r}\\\\\n\\tan \\theta &amp;= \\frac{y}{x}\n\\end{aligned}\n<\/span><\/p>\n<div class=\"edek-link-block\">\n<p><a href=\"https:\/\/www.manabi-edekeru.com\/en\/learning\/mathematics\/definition-trigonometric-functions\/\">Definition of Trigonometric Functions<\/a><\/p>\n<\/div>\n<h2 data-number=\"3\" id=\"relationship-between-trigonometric-functions-and-ratios\" class=\"anchored\"><span class=\"header-section-number\">3<\/span> Relationship Between Trigonometric Functions and Ratios<\/h2>\n<p>Comparing <a href=\"#fig-def-trigonometric-ratios\" class=\"quarto-xref\">Figure&nbsp;1<\/a> and <a href=\"#fig-def-trigonometric-funtions\" class=\"quarto-xref\">Figure&nbsp;2<\/a>, one can see that the right-angled triangle used in defining trigonometric ratios fits inside the circle that defines the trigonometric functions. Indeed, for <span class=\"math inline\">0 &lt; \\theta &lt; 90^\\circ<\/span>, there is no difference between the values expressed by trigonometric ratios and functions. However, trigonometric ratios fundamentally require the existence of a right-angled triangle. For example, at <span class=\"math inline\">\\theta = 0^\\circ<\/span> or <span class=\"math inline\">\\theta = 90^\\circ<\/span>, it is impossible to form a right-angled triangle, so trigonometric ratios cannot be defined.<\/p>\n<p>Trigonometric functions extend the trigonometric ratio concept by not requiring a right-angled triangle. Instead, they use the radius of the circle and the coordinates of the point on its circumference. Under these conditions, they define the trigonometric concepts even for angles for which trigonometric ratios cannot be defined. This extension is what trigonometric functions represent.<\/p>\n<h2 data-number=\"4\" id=\"degrees-and-radians\" class=\"anchored\"><span class=\"header-section-number\">4<\/span> Degrees and Radians<\/h2>\n<p>Trigonometric ratios often use angles with the unit \u201cdegree\u201d (<span class=\"math inline\">^\\circ<\/span>) such as <span class=\"math inline\">0^\\circ<\/span> or <span class=\"math inline\">45^\\circ<\/span>. This way of expressing angles is called the <strong>degree measure<\/strong> and is widely used in everyday life.<\/p>\n<p>On the other hand, trigonometric functions express angles using the <strong>radian measure<\/strong> rather than degrees. The radian (unit: rad) is defined as the ratio of the length of an arc of a circle to the radius of that circle (<a href=\"#def-radian\" class=\"quarto-xref\">Definition&nbsp;1<\/a>).<\/p>\n<div id=\"def-radian\" class=\"theorem definition\">\n<p><span class=\"theorem-title\"><strong>Definition 1 (Definition of an Angle Using Radians)<\/strong><\/span> <span class=\"math display\">\n\\theta = \\frac{l}{r}.\n<\/span><\/p>\n<p>Here, <span class=\"math inline\">l<\/span> is the length of the arc, and <span class=\"math inline\">r<\/span> is the radius of the circle.<\/p>\n<\/div>\n<p>In <a href=\"#def-radian\" class=\"quarto-xref\">Definition&nbsp;1<\/a>, given a circle with radius <span class=\"math inline\">r<\/span>, choose a point <span class=\"math inline\">A(r, 0)<\/span> and let point <span class=\"math inline\">B<\/span> on the circumference be reached by measuring an arc length <span class=\"math inline\">l<\/span> counterclockwise (<a href=\"#fig-def-radian\" class=\"quarto-xref\">Figure&nbsp;3<\/a>). The angle <span class=\"math inline\">\\theta<\/span> determined in this way is uniquely defined by <a href=\"#def-radian\" class=\"quarto-xref\">Definition&nbsp;1<\/a>. This angle is expressed in radians.<\/p>\n<div id=\"fig-def-radian\" class=\"quarto-float quarto-figure quarto-figure-center anchored\">\n<figure class=\"quarto-float quarto-float-fig\">\n<div aria-describedby=\"fig-def-radian-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca\">\n<img decoding=\"async\" src=\"https:\/\/www.manabi-edekeru.com\/wp-content\/uploads\/degree-and-radian.webp\">\n<\/div>\n<figcaption class=\"quarto-float-caption-bottom quarto-float-caption quarto-float-fig\" id=\"fig-def-radian-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca\">\nFigure&nbsp;3: <em>Definition of radians<\/em>\n<\/figcaption>\n<\/figure>\n<\/div>\n<p>There is a relationship between degrees and radians expressed as follows:<\/p>\n<p><span id=\"eq-relation-degree-radian\"><span class=\"math display\">\n360^\\circ \\Leftrightarrow 2 \\pi\n\\tag{1}<\/span><\/span><\/p>\n<div class=\"edek-link-block\">\n<p><a href=\"https:\/\/www.manabi-edekeru.com\/en\/learning\/mathematics\/degree-and-radian\">Degree and Radian Measures<\/a><\/p>\n<\/div>\n<h2 data-number=\"5\" id=\"sec-trig-independent-of-radii\" class=\"anchored\"><span class=\"header-section-number\">5<\/span> Trigonometric Functions Are Independent of the Radius<\/h2>\n<p>Although trigonometric functions are defined using a circle (<a href=\"#fig-def-trigonometric-funtions\" class=\"quarto-xref\">Figure&nbsp;2<\/a>), <a href=\"https:\/\/www.manabi-edekeru.com\/en\/learning\/mathematics\/trigonometric-functions-independent-of-radii\/\">their values do not depend on the radius of the circle<\/a>. No matter what radius you choose, if the angle <span class=\"math inline\">\\theta<\/span> remains the same, the trigonometric functions always have the same value.<\/p>\n<p>This justifies that when calculating the values of trigonometric functions, you may consider a circle of any convenient radius to simplify computations (<a href=\"#sec-calclulate-trigonometric-functions\" class=\"quarto-xref\">Section&nbsp;6<\/a>).<\/p>\n<h2 data-number=\"6\" id=\"sec-calclulate-trigonometric-functions\" class=\"anchored\"><span class=\"header-section-number\">6<\/span> Calculating Trigonometric Functions<\/h2>\n<p>Using the definitions of trigonometric functions shown in <a href=\"#fig-def-trigonometric-funtions\" class=\"quarto-xref\">Figure&nbsp;2<\/a>, <a href=\"https:\/\/www.manabi-edekeru.com\/en\/learning\/mathematics\/definition-trigonometric-functions\/#fig-calculate-values\">you can calculate the special values of trigonometric functions<\/a>. Let us calculate the values of trigonometric functions for <span class=\"math inline\">\\theta = \\frac{\\pi}{3}<\/span>.<\/p>\n<p>From <a href=\"#eq-relation-degree-radian\" class=\"quarto-xref\">Equation&nbsp;1<\/a>, we know that <span class=\"math inline\">2 \\pi<\/span> corresponds to <span class=\"math inline\">360^\\circ<\/span>. Since <span class=\"math inline\">\\frac{\\pi}{3}<\/span> equals <span class=\"math inline\">\\frac{1}{6}<\/span> of <span class=\"math inline\">2 \\pi<\/span>, by analogy, <span class=\"math inline\">\\frac{\\pi}{3}<\/span> corresponds to <span class=\"math inline\">60^\\circ<\/span>.<\/p>\n<p>It is well known that a right-angled triangle with an angle of <span class=\"math inline\">60^\\circ<\/span> has side ratios <span class=\"math inline\">1 : 2 : \\sqrt{3}<\/span>. Therefore, it is suitable to consider a right-angled triangle with a radius of <span class=\"math inline\">2<\/span> as in <a href=\"#fig-trig-60\" class=\"quarto-xref\">Figure&nbsp;4<\/a>. From <a href=\"#sec-trig-independent-of-radii\" class=\"quarto-xref\">Section&nbsp;5<\/a>, using a circle of radius <span class=\"math inline\">2<\/span> to calculate values of the right triangle is perfectly valid.<\/p>\n<div id=\"fig-trig-60\" class=\"quarto-float quarto-figure quarto-figure-center anchored\">\n<figure class=\"quarto-float quarto-float-fig\">\n<div aria-describedby=\"fig-trig-60-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca\">\n<img decoding=\"async\" src=\"https:\/\/www.manabi-edekeru.com\/wp-content\/uploads\/definition-trigonometric-functions-fig-calculate-values-60.webp\">\n<\/div>\n<figcaption class=\"quarto-float-caption-bottom quarto-float-caption quarto-float-fig\" id=\"fig-trig-60-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca\">\nFigure&nbsp;4: <em>Calculating trigonometric function values using radius 2<\/em>\n<\/figcaption>\n<\/figure>\n<\/div>\n<p>Following the definitions in (<a href=\"#fig-def-trigonometric-funtions\" class=\"quarto-xref\">Figure&nbsp;2<\/a>), we calculate:<\/p>\n<p><span class=\"math display\">\n\\begin{aligned}\n\\sin \\theta &amp;= \\frac{\\sqrt{3}}{2},\\\\\n\\cos \\theta &amp;= \\frac{\\sqrt{1}}{2},\\\\\n\\tan \\theta &amp;= \\frac{\\sqrt{2}}{1} = 2.\n\\end{aligned}\n<\/span><\/p>\n<p>In this example, we use a circle of radius 2. For other angles such as <span class=\"math inline\">\\theta = \\frac{\\pi}{4}<\/span>, it is easier to consider an isosceles right triangle and use a circle of radius <span class=\"math inline\">\\sqrt{2}<\/span>.<\/p>\n<div class=\"edek-link-block\">\n<p><a href=\"https:\/\/www.manabi-edekeru.com\/en\/learning\/mathematics\/definition-trigonometric-functions\/#fig-calculate-values\">How to Calculate Special Values Using Definitions<\/a><\/p>\n<\/div>\n<h2 data-number=\"7\" id=\"trigonometric-identities\" class=\"anchored\"><span class=\"header-section-number\">7<\/span> Trigonometric Identities<\/h2>\n<p>There are many trigonometric identities, but most can be proved or derived easily by using definitions or addition formulas. Understanding the methods to prove these identities significantly reduces the number you need to memorise.<\/p>\n<p>Rather than rote learning, <a href=\"https:\/\/www.manabi-edekeru.com\/en\/learning\/mathematics\/derive-trigonometric-identities-instead-of-memorising\/\">comprehending and being able to prove the identities yourself<\/a> deepens your understanding of trigonometric functions, enhances your ability to apply them, and leads to better long-term retention.<\/p>\n<p><a href=\"https:\/\/www.manabi-edekeru.com\/en\/learning\/mathematics\/trigonometric-identities\/\">Trigonometric identities<\/a> are classified into two groups based on their proof method: (1) those proved using the definition, and (2) those proved using addition formulas. I recommend understanding the identities based on this classification. For a comprehensive summary, please see the articles below:<\/p>\n<p>Identities proved using the definition include <a href=\"https:\/\/www.manabi-edekeru.com\/en\/learning\/mathematics\/basic-trigonometric-identities\/\">basic trigonometric identities<\/a>, <a href=\"https:\/\/www.manabi-edekeru.com\/en\/learning\/mathematics\/trigonometric-identities-negative-theta\/\">negative angle trigonometric identities<\/a>, <a href=\"https:\/\/www.manabi-edekeru.com\/en\/learning\/mathematics\/trigonometric-identities-complementary-angles\/\">complementary angle identities<\/a>, and <a href=\"https:\/\/www.manabi-edekeru.com\/en\/learning\/mathematics\/trigonometric-identities-supplementary-angles\/\">supplementary angle identities<\/a>. Please check the following articles for the proofs of each identitiy:<\/p>\n<ul>\n<li><a href=\"https:\/\/www.manabi-edekeru.com\/en\/learning\/mathematics\/basic-trigonometric-identities\/\">Proofs of basic trigonometric identities<\/a><\/li>\n<li><a href=\"https:\/\/www.manabi-edekeru.com\/en\/learning\/mathematics\/proof-trigonometric-identities-negative-theta\/\">Proofs of negative angle identities<\/a><\/li>\n<li><a href=\"https:\/\/www.manabi-edekeru.com\/en\/learning\/mathematics\/proof-trigonometric-identities-complementary-angles\/\">Proofs of complementary angle identities<\/a><\/li>\n<li><a href=\"https:\/\/www.manabi-edekeru.com\/en\/learning\/mathematics\/poof-trigonometric-identities-supplementary-angles\/\">Proofs of supplemetary angle identities<\/a><\/li>\n<\/ul>\n<p>Along with the definition, it is advisable to memorise the addition theorems. To enhance your application skills with trigonometric functions, you should understand the proofs of the addition theorems. However, recalling these theorems through proof can be quite tiresome. Therefore, ensure you remember both the definition and the addition theorems.<\/p>\n<h2 data-number=\"8\" id=\"graphs-of-trigonometric-functions\" class=\"anchored\"><span class=\"header-section-number\">8<\/span> Graphs of Trigonometric Functions<\/h2>\n<p>When studying functions, we often rely on their <strong>graphs<\/strong>. Exploring the graphs of trigonometric functions helps deepen your understanding of their properties.<\/p>\n<h3 data-number=\"8.1\" id=\"periodicity-of-trigonometric-functions\" class=\"anchored\"><span class=\"header-section-number\">8.1<\/span> Periodicity of Trigonometric Functions<\/h3>\n<p>As shown in <a href=\"#fig-trig-graphs-sin-cos-tan\" class=\"quarto-xref\">Figure&nbsp;5<\/a>, the graphs of trigonometric functions exhibit a <strong>periodic<\/strong> pattern. Both <span class=\"math inline\">y = \\sin x<\/span> and <span class=\"math inline\">y = \\cos x<\/span> have a fundamental period of <span class=\"math inline\">2 \\pi<\/span>, whereas <span class=\"math inline\">y = \\tan x<\/span> has a period of <span class=\"math inline\">\\pi<\/span>. A period refers to the change in <span class=\"math inline\">x<\/span> that brings the function back to the same value. More precisely, a constant <span class=\"math inline\">p<\/span> is a period of the function <span class=\"math inline\">f<\/span> if:<\/p>\n<p><span class=\"math display\">\nf(x) = f(x + p)\n<\/span><\/p>\n<p>This equation means that for any <span class=\"math inline\">x<\/span> in the domain, shifting by <span class=\"math inline\">p<\/span> results in the same <span class=\"math inline\">y<\/span> value. For example, in <a href=\"#fig-trig-graphs-sin-cos-tan\" class=\"quarto-xref\">Figure&nbsp;5<\/a> (a), at <span class=\"math inline\">x = 0<\/span>, <span class=\"math inline\">y = 0<\/span>, and at <span class=\"math inline\">x = 2 \\pi<\/span>, which is <span class=\"math inline\">2 \\pi<\/span> shifted from <span class=\"math inline\">0<\/span>, <span class=\"math inline\">y<\/span> also equals <span class=\"math inline\">0<\/span>.<\/p>\n<p>Although <span class=\"math inline\">y = 0<\/span> also occurs at <span class=\"math inline\">x = \\pi<\/span>, shifting <span class=\"math inline\">x = \\frac{\\pi}{2}<\/span> (the peak) by <span class=\"math inline\">\\pi<\/span> leads to <span class=\"math inline\">x = \\frac{3 \\pi}{2}<\/span>, where the function value corresponds to a trough rather than the same <span class=\"math inline\">y<\/span> value. Therefore, <span class=\"math inline\">\\pi<\/span> is not a period of <span class=\"math inline\">y = \\sin x<\/span>.<\/p>\n<p>Similarly, <span class=\"math inline\">y = \\cos x<\/span> has period <span class=\"math inline\">2 \\pi<\/span>, showing oscillations similar to <span class=\"math inline\">\\sin x<\/span>.<\/p>\n<p>In contrast, <span class=\"math inline\">y = \\tan x<\/span> has a period of <span class=\"math inline\">\\pi<\/span>, and it is undefined at points like <span class=\"math inline\">x = \\frac{\\pi}{2}<\/span> or <span class=\"math inline\">x = -\\frac{\\pi}{2}<\/span>.<\/p>\n<div id=\"fig-trig-graphs-sin-cos-tan\" class=\"quarto-float quarto-figure quarto-figure-center anchored\">\n<figure class=\"quarto-float quarto-float-fig\">\n<div aria-describedby=\"fig-trig-graphs-sin-cos-tan-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca\">\n<div id=\"cell-trigonometric-functions-fig-trig-graphs-sin-cos-tan\" class=\"cell\" data-execution_count=\"3\">\n<div class=\"cell-output cell-output-display\">\n<div>\n<figure>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.manabi-edekeru.com\/wp-content\/uploads\/trigonometric-functions-fig-trig-graphs-sin-cos-tan-output-1.png\" id=\"trigonometric-functions-fig-trig-graphs-sin-cos-tan\" width=\"1035\" height=\"737\"><\/p>\n<\/figure>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<figcaption class=\"quarto-float-caption-bottom quarto-float-caption quarto-float-fig\" id=\"fig-trig-graphs-sin-cos-tan-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca\">\nFigure&nbsp;5: <em>Period of trigonometric functions<\/em><br>\n<em>Purple bar: period. Yellow bar: amplitude.<\/em>\n<\/figcaption>\n<\/figure>\n<\/div>\n<h3 data-number=\"8.2\" id=\"amplitude-maximum-and-minimum-values-of-trigonometric-functions\" class=\"anchored\"><span class=\"header-section-number\">8.2<\/span> Amplitude, Maximum and Minimum Values of Trigonometric Functions<\/h3>\n<p>\u201cAmplitude\u201d refers to the <strong>height of the wave from its central position to its peak<\/strong>. In figures <a href=\"#fig-trig-graphs-sin-cos-tan\" class=\"quarto-xref\">Figure&nbsp;5<\/a> (a) and (b), the yellow bar represents the amplitude. The centre of the wave height corresponds to the <span class=\"math inline\">x<\/span>-axis. Both <span class=\"math inline\">y = \\sin x<\/span> and <span class=\"math inline\">y = \\cos x<\/span> have an amplitude of <span class=\"math inline\">1<\/span>.<\/p>\n<p>As shown in <a href=\"#fig-trig-graphs-sin-cos-tan\" class=\"quarto-xref\">Figure&nbsp;5<\/a>, the amplitude essentially equals the maximum value. Therefore, the maximum and minimum values of <span class=\"math inline\">y = \\sin x<\/span> and <span class=\"math inline\">y = \\cos x<\/span> are <span class=\"math inline\">1<\/span> and <span class=\"math inline\">-1<\/span> respectively. On the other hand, <span class=\"math inline\">y = \\tan x<\/span> takes any real value from <span class=\"math inline\">&#8211; \\infty<\/span> to <span class=\"math inline\">\\infty<\/span>.<\/p>\n<h3 data-number=\"8.3\" id=\"domain-and-range-of-trigonometric-functions\" class=\"anchored\"><span class=\"header-section-number\">8.3<\/span> Domain and Range of Trigonometric Functions<\/h3>\n<p>A function relates each input value to exactly one output value. The set of all possible input values is called the \u201cdomain,\u201d and the set of all possible output values is called the \u201crange.\u201d<\/p>\n<p>For trigonometric functions, the domain consists of all possible angles. As shown in <a href=\"#fig-trig-graphs-sin-cos-tan\" class=\"quarto-xref\">Figure&nbsp;5<\/a>, we can consider the domain of <span class=\"math inline\">y = \\sin x<\/span> and <span class=\"math inline\">y = \\cos x<\/span> as <strong>all real numbers<\/strong>. In contrast, <span class=\"math inline\">y = \\tan x<\/span> is not defined at <span class=\"math inline\">x = \\frac{\\pi}{2} + n \\pi<\/span>, where <span class=\"math inline\">n<\/span> is any integer. Here, <span class=\"math inline\">\\mathbb{Z}<\/span> represents the set of all integers (<a href=\"#tbl-domain-range\" class=\"quarto-xref\">Table&nbsp;1<\/a>).<\/p>\n<p>Since <span class=\"math inline\">y = \\sin x<\/span> and <span class=\"math inline\">y = \\cos x<\/span> take values between <span class=\"math inline\">-1<\/span> and <span class=\"math inline\">1<\/span>, their range is <span class=\"math inline\">-1 \\le y \\le 1<\/span>. Meanwhile, <span class=\"math inline\">y = \\tan x<\/span> can take any real number, so its range is <span class=\"math inline\">-\\infty &lt; y &lt; \\infty<\/span> (<a href=\"#tbl-domain-range\" class=\"quarto-xref\">Table&nbsp;1<\/a>).<\/p>\n<div id=\"tbl-domain-range\" class=\"quarto-float quarto-figure quarto-figure-center anchored\">\n<figure class=\"quarto-float quarto-float-tbl\">\n<figcaption class=\"quarto-float-caption-top quarto-float-caption quarto-float-tbl\" id=\"tbl-domain-range-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca\">\nTable&nbsp;1: Domain and range of trigonometric functions<br>\n\n<\/figcaption>\n<div aria-describedby=\"tbl-domain-range-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca\">\n<table class=\"caption-top\">\n<colgroup>\n<col style=\"width: 11%\">\n<col style=\"width: 66%\">\n<col style=\"width: 21%\">\n<\/colgroup>\n<thead>\n<tr class=\"header\">\n<th style=\"text-align: center;\">Function<\/th>\n<th style=\"text-align: center;\">Domain<\/th>\n<th style=\"text-align: center;\">Range<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr class=\"odd\">\n<td style=\"text-align: center;\"><span class=\"math inline\">y = \\sin x<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"math inline\">-\\infty &lt; x &lt; \\infty<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"math inline\">-1 \\le y \\le 1<\/span><\/td>\n<\/tr>\n<tr class=\"even\">\n<td style=\"text-align: center;\"><span class=\"math inline\">y = \\cos x<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"math inline\">-\\infty &lt; x &lt; \\infty<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"math inline\">-1 \\le y \\le 1<\/span><\/td>\n<\/tr>\n<tr class=\"odd\">\n<td style=\"text-align: center;\"><span class=\"math inline\">y = \\tan x<\/span><\/td>\n<td style=\"text-align: center;\">All real numbers except <span class=\"math inline\">x = \\frac{\\pi}{2} + n\\pi, n \\in \\mathbb{Z}<\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"math inline\">-\\infty &lt; y &lt; \\infty<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/figure>\n<\/div>\n<h3 data-number=\"8.4\" id=\"even-and-odd-functions-in-trigonometry\" class=\"anchored\"><span class=\"header-section-number\">8.4<\/span> Even and Odd Functions in Trigonometry<\/h3>\n<p>A notable feature of trigonometric functions, <span class=\"math inline\">y = \\sin x<\/span>, <span class=\"math inline\">y = \\cos x<\/span>, <span class=\"math inline\">y = \\tan x<\/span>, is that they are even or odd functions. Simply put, these functions satisfy the following conditions:<\/p>\n<ul>\n<li><strong>Even function<\/strong>: The graph is symmetrical about the <span class=\"math inline\">y<\/span>-axis.<\/li>\n<li><strong>Odd function<\/strong>: The graph is symmetrical about the origin.<\/li>\n<\/ul>\n<p>As shown in <a href=\"#fig-trig-graphs-sin-cos-tan\" class=\"quarto-xref\">Figure&nbsp;5<\/a>, the graph of <span class=\"math inline\">y = \\sin x<\/span> is symmetric about the origin (rotating it <span class=\"math inline\">180^\\circ<\/span> around the origin results in the same graph), so it is an odd function. Similarly, <span class=\"math inline\">y = \\tan x<\/span> is also an odd function. Meanwhile, <span class=\"math inline\">y = \\cos x<\/span> is symmetric about the <span class=\"math inline\">y<\/span>-axis (folding along the <span class=\"math inline\">y<\/span>-axis matches the graph onto itself), so it is an even function.<\/p>\n<h2 data-number=\"9\" id=\"parallel-translation-stretching-and-shrinking-of-trigonometric-graphs\" class=\"anchored\"><span class=\"header-section-number\">9<\/span> Parallel Translation, Stretching and Shrinking of Trigonometric Graphs<\/h2>\n<p>Using the graphs shown in <a href=\"#fig-trig-graphs-sin-cos-tan\" class=\"quarto-xref\">Figure&nbsp;5<\/a> as a basis, we sometimes consider graphs that have been translated (shifted) or stretched\/shrunk.<\/p>\n<h3 data-number=\"9.1\" id=\"parallel-translation-of-trigonometric-graphs\" class=\"anchored\"><span class=\"header-section-number\">9.1<\/span> Parallel Translation of Trigonometric Graphs<\/h3>\n<p>Generally, the graph of a function <span class=\"math inline\">y = f(x)<\/span> translated by <span class=\"math inline\">p<\/span> units in the <span class=\"math inline\">x<\/span>-direction and <span class=\"math inline\">q<\/span> units in the <span class=\"math inline\">y<\/span>-direction is expressed as:<\/p>\n<p><span class=\"math display\">\ny &#8211; q = f(x &#8211; p).\n<\/span><\/p>\n<p>This function is obtained by replacing <span class=\"math inline\">x<\/span> by <span class=\"math inline\">x &#8211; p<\/span> and <span class=\"math inline\">y<\/span> by <span class=\"math inline\">y &#8211; q<\/span> in <span class=\"math inline\">y = f(x)<\/span>.<\/p>\n<p>Therefore, the functions representing the graphs of these trigonometric functions translated by <span class=\"math inline\">p<\/span> horizontally and <span class=\"math inline\">q<\/span> vertically are:<\/p>\n<p><span id=\"eq-trig-parallel-translation\"><span class=\"math display\">\n\\begin{aligned}\ny &amp;= \\sin x \\quad \\rightarrow \\quad y &#8211; q = \\sin (x &#8211; p), \\\\\ny &amp;= \\cos x \\quad \\rightarrow \\quad y &#8211; q = \\cos (x &#8211; p), \\\\\ny &amp;= \\tan x \\quad \\rightarrow \\quad y &#8211; q = \\tan (x &#8211; p).\n\\end{aligned}\n\\tag{2}<\/span><\/span><\/p>\n<p>In <a href=\"#fig-trig-graphs-pt-sin-cos-tan\" class=\"quarto-xref\">Figure&nbsp;6<\/a>, examples of such translated graphs are shown. The purple dashed lines represent the original graphs, such as <span class=\"math inline\">y = \\sin x<\/span>, while the translated graphs are shown with blue solid lines.<\/p>\n<p>For example, in <a href=\"#fig-trig-graphs-pt-sin-cos-tan\" class=\"quarto-xref\">Figure&nbsp;6<\/a> (a), the graph of <span class=\"math inline\">y = \\sin x<\/span> is translated to <span class=\"math inline\">y = \\sin \\left( x &#8211; \\frac{\\pi}{2} \\right)<\/span>. Notice the purple dot on the dashed line graph moves to the blue dot on the solid line graph.<\/p>\n<p>Comparing with <a href=\"#eq-trig-parallel-translation\" class=\"quarto-xref\">Equation&nbsp;2<\/a>, this example shifts the graph of <span class=\"math inline\">y = \\sin x<\/span> by <span class=\"math inline\">\\frac{\\pi}{2}<\/span> along the <span class=\"math inline\">x<\/span>-axis.<\/p>\n<p>In <a href=\"#fig-trig-graphs-pt-sin-cos-tan\" class=\"quarto-xref\">Figure&nbsp;6<\/a> (c), the graph of <span class=\"math inline\">y = \\tan x<\/span> is translated by <span class=\"math inline\">\\frac{\\pi}{2}<\/span> horizontally and by <span class=\"math inline\">1<\/span> vertically.<\/p>\n<div id=\"fig-trig-graphs-pt-sin-cos-tan\" class=\"quarto-float quarto-figure quarto-figure-center anchored\">\n<figure class=\"quarto-float quarto-float-fig\">\n<div aria-describedby=\"fig-trig-graphs-pt-sin-cos-tan-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca\">\n<div id=\"cell-trigonometric-functions-fig-trig-graphs-pt-sin-cos-tan\" class=\"cell\" data-execution_count=\"4\">\n<div class=\"cell-output cell-output-display\">\n<div>\n<figure>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.manabi-edekeru.com\/wp-content\/uploads\/trigonometric-functions-fig-trig-graphs-pt-sin-cos-tan-output-1.png\" id=\"trigonometric-functions-fig-trig-graphs-pt-sin-cos-tan\" width=\"1035\" height=\"741\"><\/p>\n<\/figure>\n<\/div>\n<\/div>\n<\/div>\n<p><em>Parallel translation and scaling of trigonometric Functions<\/em><br>\n<\/p>\n<\/div>\n<figcaption class=\"quarto-float-caption-bottom quarto-float-caption quarto-float-fig\" id=\"fig-trig-graphs-pt-sin-cos-tan-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca\">\nFigure&nbsp;6: <em>The purple dashed lines represent the graphs of <span class=\"math inline\">y = \\sin x<\/span>, <span class=\"math inline\">y = \\cos x<\/span>, and <span class=\"math inline\">y = \\tan x<\/span>. The blue solid lines show the graphs after a parallel translation. Because of this translation, the purple dots on the dashed lines move to the dots on the solid lines.<\/em>\n<\/figcaption>\n<\/figure>\n<\/div>\n<h3 data-number=\"9.2\" id=\"scaling-of-trigonometric-graphs\" class=\"anchored\"><span class=\"header-section-number\">9.2<\/span> Scaling of Trigonometric Graphs<\/h3>\n<p>You can scale the graph of a trigonometric function along the <span class=\"math inline\">y<\/span>-axis or the <span class=\"math inline\">x<\/span>-axis. For example, the following equation represents scaling the graph of <span class=\"math inline\">y = \\cos x<\/span> by a factor of <span class=\"math inline\">a<\/span> along the <span class=\"math inline\">y<\/span>-axis:<\/p>\n<p><span id=\"eq-trig-scaling-y\"><span class=\"math display\">\ny = a \\cos x.\n\\tag{3}<\/span><\/span><\/p>\n<p>In figure <a href=\"#fig-trig-graphs-pt-sin-cos-tan\" class=\"quarto-xref\">Figure&nbsp;6<\/a> (b), the graph of <span class=\"math inline\">y = \\cos x<\/span> is scaled by a factor of <span class=\"math inline\">2<\/span> along the <span class=\"math inline\">y<\/span>-axis.<\/p>\n<p>On the other hand, the following equation scales the graph of <span class=\"math inline\">y = \\cos x<\/span> along the <span class=\"math inline\">x<\/span>-axis by a factor of <span class=\"math inline\">\\frac{1}{b}<\/span>:<\/p>\n<p><span id=\"eq-trig-scaling-x\"><span class=\"math display\">\ny = \\cos (bx).\n\\tag{4}<\/span><\/span><\/p>\n<p>This means that when <span class=\"math inline\">b=2<\/span>, as in <span class=\"math inline\">y = \\cos (2x)<\/span>, the graph is compressed to half its width along the <span class=\"math inline\">x<\/span>-axis. As a result, the number of wave peaks between <span class=\"math inline\">-2 \\pi \\le x \\le 2 \\pi<\/span> increases (see figure <a href=\"#fig-trig-graphs-pt-sin-cos-tan\" class=\"quarto-xref\">Figure&nbsp;6<\/a> (b)).<\/p>\n<h3 data-number=\"9.3\" id=\"summary-of-parallel-translation-and-scaling-of-trigonometric-graphs\" class=\"anchored\"><span class=\"header-section-number\">9.3<\/span> Summary of Parallel Translation and Scaling of Trigonometric Graphs<\/h3>\n<p>Combining equations <a href=\"#eq-trig-parallel-translation\" class=\"quarto-xref\">Equation&nbsp;2<\/a>, <a href=\"#eq-trig-scaling-y\" class=\"quarto-xref\">Equation&nbsp;3<\/a>, and <a href=\"#eq-trig-scaling-x\" class=\"quarto-xref\">Equation&nbsp;4<\/a>, the following equation expresses all these transformations of trigonometric graphs\u2014parallel translation, scaling, or both\u2014(shown here for <span class=\"math inline\">y = \\cos x<\/span>, but the same applies to other functions):<\/p>\n<p><span class=\"math display\">\n\\begin{aligned}\ny &#8211; q &amp;= a \\cos (b(x &#8211; p))\\\\\n&amp;\\Leftrightarrow\\\\\ny &amp;= a \\cos (b(x &#8211; p)) + q.\n\\end{aligned}\n<\/span><\/p>\n<p>The graph drawn b y this function has the following characteristics:<\/p>\n<ul>\n<li>Amplitude: <span class=\"math inline\">\\vert a \\vert<\/span><\/li>\n<li>Period: <span class=\"math inline\">\\frac{2 \\pi}{\\vert b \\vert}<\/span><\/li>\n<li>Horizontal (phase) shift: <span class=\"math inline\">p<\/span><\/li>\n<li>Vertical shift: <span class=\"math inline\">q<\/span><\/li>\n<\/ul>\n<p>In figure <a href=\"#fig-trig-graphs-transformation\" class=\"quarto-xref\">Figure&nbsp;7<\/a>, the graph of <span class=\"math inline\">y = 2 \\cos \\left(\\frac{1}{2} \\left(x &#8211; \\frac{\\pi}{2} \\right)\\right) + 1<\/span> is plotted. I encourage you to verify the amplitude, period, and both <span class=\"math inline\">x<\/span>-axis and <span class=\"math inline\">y<\/span>-axis shifts using this example.<\/p>\n<div id=\"fig-trig-graphs-transformation\" class=\"quarto-float quarto-figure quarto-figure-center anchored\">\n<figure class=\"quarto-float quarto-float-fig\">\n<div aria-describedby=\"fig-trig-graphs-transformation-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca\">\n<div id=\"cell-trigonometric-functions-fig-trig-graphs-transformation\" class=\"cell\" data-execution_count=\"5\">\n<div class=\"cell-output cell-output-display\">\n<div>\n<figure>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.manabi-edekeru.com\/wp-content\/uploads\/trigonometric-functions-fig-trig-graphs-transformation-output-1.png\" id=\"trigonometric-functions-fig-trig-graphs-transformation\" width=\"898\" height=\"643\"><\/p>\n<\/figure>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<figcaption class=\"quarto-float-caption-bottom quarto-float-caption quarto-float-fig\" id=\"fig-trig-graphs-transformation-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca\">\nFigure&nbsp;7: <em>Transformation of the graph of trigonometric functions<\/em><br>\n<em>The purple dashed line represents <span class=\"math inline\">y = \\cos x<\/span>. The blue solid line shows <span class=\"math inline\">y = 2 \\cos \\left(\\frac{1}{2} \\left(x &#8211; \\frac{\\pi}{2} \\right)\\right) + 1<\/span>. The purple dots on the dashed line move to the dots on the solid line.<\/em>\n<\/figcaption>\n<\/figure>\n<\/div>\n<h2 data-number=\"10\" id=\"conclusion\" class=\"anchored\"><span class=\"header-section-number\">10<\/span> Conclusion<\/h2>\n<p>In this article, we have broadly covered the fundamental concepts of trigonometric functions. You can access more detailed explanations through the links provided within the text. Understanding the proofs of the formulas for trigonometric functions significantly reduces what you need to memorise. Use what you have learned here as a foundation to deepen your understanding of trigonometric functions further.<\/p>\n\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<ul>\n<li>The overview of trigonometric functions<\/li>\n<li>Definition, formula, graph transformations<\/li>\n<\/ul>\n","protected":false},"author":3,"featured_media":12796,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_locale":"en_US","_original_post":"https:\/\/www.manabi-edekeru.com\/?p=12017","iawp_total_views":0,"footnotes":""},"categories":[3],"tags":[],"free_or_paid":[83],"level":[55],"post_language":[49],"resource_type":[59],"topic":[82,51],"class_list":["post-12017","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics","free_or_paid-free","level-intermediate","post_language-english","resource_type-reading","topic-trigonometry","topic-mathematics","en-US","entry"],"yoast_head":"<!-- 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