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378 lines
19 KiB
HTML
378 lines
19 KiB
HTML
<!DOCTYPE html>
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<html lang="en"><!-- #BeginTemplate "/Templates/Main.dwt" --><!-- DW6 -->
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<!-- Mirrored from www.mathsisfun.com/geometry/unit-circle.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:37:40 GMT -->
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Unit Circle</h1>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/unit-circle.svg" alt="unit circle center at (0,0)" height="222" width="242"></p>
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<p> </p>
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<p class="larger">The "Unit Circle" is a circle with a radius of 1.</p>
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<p>Being so simple, it is a great way to learn and talk about lengths and angles.</p>
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<p>The center is put on a graph where the x axis and y axis cross, so we get this neat arrangement here.</p>
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<p> </p>
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<div style="clear:both"></div>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/unit-circle-sin-cos-tan.svg" alt="unit circle center at (0,0)" height="220" width="277"></p>
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<h2>Sine, Cosine and Tangent</h2>
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<p>Because the radius is 1, we can directly measure <a href="../sine-cosine-tangent.html">sine, cosine and tangent</a>.</p>
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<div style="clear:both"></div>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/unit-circle-angle0.svg" alt="unit circle center angle 0" height="219" width="256"></p>
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<p>What happens when the angle, θ, is 0°?</p>
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<p class="so">cos 0° = 1, sin 0° = 0 and tan 0° = 0</p>
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<div style="clear:both"></div>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/unit-circle-angle90.svg" alt="unit circle center angle 90" height="219" width="215"></p>
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<p>What happens when θ is 90°?</p>
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<p class="so">cos 90° = 0, sin 90° = 1 and tan 90° is undefined</p>
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<div style="clear:both"></div>
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<h2>Try It Yourself!</h2>
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<p>Have a try! Move the mouse around to see how different angles (in <a href="radians.html">radians</a> or <a href="degrees.html">degrees</a>) affect sine, cosine and tangent</p>
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<div class="script" style="height: 560px;">
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../algebra/images/circle-triangle.js
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</div>
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<p>The "sides" can be positive or negative according to the rules of <a href="../data/cartesian-coordinates.html">Cartesian coordinates</a>. This makes the sine, cosine and tangent change between positive and negative values also.</p>
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<p> </p>
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<p class="center larger">Also try the <a href="../algebra/trig-interactive-unit-circle.html">Interactive Unit Circle</a>.</p>
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<div style="clear:both"></div>
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<p> </p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/unit-circle-xy.svg" alt="unit circle center at (0,0)" height="289" width="296"></p>
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<h2>Pythagoras</h2>
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<p><a href="../pythagoras.html">Pythagoras' Theorem</a> says that for a right angled triangle, the square of the long side equals the sum of the squares of the other two sides:</p>
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<p class="center larger">x<sup>2</sup> + y<sup>2</sup> = 1<sup>2</sup></p>
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<p>But 1<sup>2</sup> is just 1, so:</p>
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<p class="center large"><span class="larger">x<sup>2</sup> + y<sup>2</sup> = 1 </span><i><br>
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equation of the unit circle</i></p>
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<p>Also, since x=cos and y=sin, we get:</p>
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<p class="center large"><span class="larger">(cos(θ))<sup>2</sup> + (sin(θ))<sup>2</sup> = 1</span><br>
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<i>a useful "identity"</i></p>
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<h2>Important Angles: 30<i>°</i>, 45<i>°</i> and 60<i>°</i></h2>
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<p>You should try to <b>remember</b> sin, cos and tan for the angles 30<i>°</i>, 45<i>°</i> and 60<i>°</i><b>.</b></p>
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<p>Yes, yes, it is a pain to have to remember things, but it will make life easier when you know them, not just in exams, but other times when you need to do quick estimates, etc.</p>
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<p class="center"><b>These are the values you should remember!</b></p>
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<div class="beach">
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<table align="center" cellspacing="0" cellpadding="5" bordercolor="#0066CC" border="1">
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<tbody>
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<tr style="text-align:center;">
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<th width="100">Angle</th>
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<th width="100">Cos</th>
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<th width="100">Sin</th>
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<th width="100"><i>Tan=Sin/Cos</i></th>
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</tr>
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<tr style="text-align:center;">
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<th width="100">30<i>°</i></th>
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<td style="width:100px;"><span class="intbl"><em>√3</em><strong>2</strong></span></td>
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<td><span class="intbl"><em>1</em><strong>2</strong></span></td>
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<td style="width:100px;"><span class="intbl">
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<em>1</em>
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<strong>√3</strong>
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</span>
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= <span class="intbl">
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<em>√3</em>
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<strong>3</strong>
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</span></td>
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</tr>
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<tr style="text-align:center;">
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<th width="100">45<i>°</i></th>
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<td style="width:100px;"><span class="intbl"><em>√2</em><strong>2</strong></span></td>
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<td><span class="intbl"><em>√2</em><strong>2</strong></span></td>
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<td style="width:100px;">1</td>
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</tr>
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<tr style="text-align:center;">
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<th width="100">60<i>°</i></th>
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<td style="width:100px;"><span class="intbl"><em>1</em><strong>2</strong></span></td>
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<td><span class="intbl"><em>√3</em><strong>2</strong></span></td>
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<td style="width:100px;">√3</td>
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</tr>
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</tbody></table><br>
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</div>
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<div class="center80">
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<h3>How To Remember?</h3>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/circle-unit-123.svg" alt="unit circle 123" height="182" width="209"></p>
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<p>To help you remember, cos goes <b>"3,2,1"</b></p>
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<p class="large"> cos(30<i>°</i>) = <span class="intbl"><em>√<span class="hilite"><b>3</b></span></em><strong>2</strong></span></p>
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<p class="large"> cos(45<i>°</i>) = <span class="intbl"><em>√<span class="hilite"><b>2</b></span></em><strong>2</strong></span></p>
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<p class="large"> cos(60<i>°</i>) = <span class="intbl"><em>√<span class="hilite"><b>1</b></span></em><strong>2</strong></span> = <span class="intbl"><em>1</em><strong>2</strong></span></p>
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<p> </p>
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<p>And, sin goes <b>"1,2,3"</b> :</p>
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<p class="large"> sin(30<i>°</i>) = <span class="intbl"><em>√<span class="hilite"><b>1</b></span></em><strong>2</strong></span> = <span class="intbl"><em>1</em><strong>2</strong></span> <span style="font-size: 80%">(because √1 = 1)</span></p>
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<p class="large"> sin(45<i>°</i>) = <span class="intbl"><em>√<span class="hilite"><b>2</b></span></em><strong>2</strong></span></p>
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<p class="large"> sin(60<i>°</i>) = <span class="intbl"><em>√<span class="hilite"><b>3</b></span></em><strong>2</strong></span></p>
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<h2>Just 3 Numbers</h2>
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<p>In fact, knowing 3 numbers is enough: <span class="intbl"><em>1</em><strong>2</strong></span> , <span class="intbl"><em>√2</em><strong>2</strong></span> and <span class="intbl"><em>√3</em><strong>2</strong></span></p>
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<p>Because they work for both <b>cos</b> and <b>sin</b>:</p>
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<p class="center"><img src="images/unit-circle-cos.svg" alt="unit circle cos 1/2, root2/2, root3/2" height="217" width="211"> <img src="images/unit-circle-sin.svg" alt="unit circle cos 1/2, root2/2, root3/2" height="217" width="211"></p>
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<p>
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Your hand can help you remember:
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</p>
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<p class="center"><img src="images/unit-circle-hand.svg" alt="unit circle cos 1/2, root2/2, root3/2" height="" width=""> </p>
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<p>
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For example there are 3 fingers above 30°, so cos(30°) = <span class="intbl"><em><span style="font-size:120%;">√</span><span class="overline">3</span></em><strong>2</strong></span>
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</p>
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</div>
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<h2>What about tan?</h2>
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<p>Well, <b>tan = sin/cos</b>, so we can calculate it like this:</p>
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<p class="large">tan(30°) =<span class="intbl"><em>sin(30°)</em><strong>cos(30°)</strong></span><b> = </b><span class="intbl"><em>1/2</em><strong>√3/2</strong></span> = <span class="intbl"><em>1</em><strong>√3</strong></span> = <b><span class="intbl"><em>√3</em><strong>3</strong></span></b> *</p>
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<p class="large">tan(45°) =<span class="intbl"><em>sin(45°)</em><strong>cos(45°)</strong></span><b> = </b><span class="intbl"><em>√2/2</em><strong>√2/2</strong></span> =<b> 1 </b></p>
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<p class="large">tan(60°) =<span class="intbl"><em>sin(60°)</em><strong>cos(60°)</strong></span><b> = </b><span class="intbl"><em>√3/2</em><strong>1/2</strong></span> =<b> √3 </b></p>
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<p>* Note: writing <span class="intbl"><em>1</em><strong>√3</strong></span> <b>may cost you marks</b> so use <b><span class="intbl"><em>√3</em><strong>3</strong></span></b> instead (see <a href="../algebra/rationalize-denominator.html">Rational Denominators</a> to learn more). <b><span class="intbl"></span></b></p>
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<h2>Quick Sketch</h2>
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<p>Another way to help you remember 30° and 60° is to make a quick sketch:</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td>Draw a triangle with side lengths of 2</td>
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<td> </td>
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<td><img src="images/triangle-30-60-sketch-a.gif" alt="triangle 60 60 with sides of 2" height="113" width="110"></td>
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</tr>
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<tr>
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<td>
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<p>Cut in half. <a href="../pythagoras.html">Pythagoras</a> says the new side is √3</p>
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<div class="so">1<sup>2</sup> + (√3)<sup>2</sup> = 2<sup>2</sup></div>
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<div class="so">1 + 3 = 4</div></td>
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<td> </td>
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<td><img src="images/triangle-30-60-sketch-b.gif" alt="triangle 30 60 with sides of 1, 2, root3" height="112" width="110"></td>
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</tr>
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<tr>
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<td>Then use <a href="../algebra/sohcahtoa.html">sohcahtoa</a> for sin, cos or tan</td>
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<td> </td>
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<td><img src="images/triangle-30-60-sketch-c.gif" alt="triangle 30 60 with sides of 1, 2, root3" height="110" width="110"></td>
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</tr>
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</tbody></table>
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<div class="example">
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<h3>Example: sin(30°)</h3>
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<p>Sine: <b>soh</b>cahtoa</p>
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<div class="so">sine is opposite divided by hypotenuse</div>
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<div class="so">sin(30°) = <span class="intbl">
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<em>opposite</em>
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<strong>hypotenuse</strong>
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</span> = <span class="intbl">
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<em>1</em>
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<strong>2</strong>
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</span> </div>
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</div>
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<p> </p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/quadrants.gif" alt="quadrants (+,+) (-,+) (-,-) and (+,-) going counterclockwise" height="190" width="225"></p>
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<h2>The Whole Circle</h2>
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<p>For the whole circle we need values in <a href="../algebra/trig-four-quadrants.html">every quadrant</a>, with the correct plus or minus sign as per <a href="../data/cartesian-coordinates.html">Cartesian Coordinates</a>:</p>
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<p> </p>
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<p>Note that <b>cos</b> is first and <b>sin</b> is second, so it goes <b>(cos, sin)</b>:</p>
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<div style="clear:both"></div>
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<p class="center"><img src="images/circle-unit-304560.svg" alt="Unit Circle Degrees" height="397" width="496"></p>
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<p class="center"><a href="images/circle-unit.pdf">Save as PDF</a></p>
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<div class="example">
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<h3>Example: What is cos(330°) ?</h3>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/unit-circle-330.svg" alt="unit circle 330" height="158" width="182"></p>
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<p> </p>
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<p>Make a sketch like this, and we can see it is the "long" value: <span class="intbl"><em>√3</em><strong>2</strong></span></p>
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</div>
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<p align="left">And this is the same Unit Circle in <b>radians</b>.</p>
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<p class="center"><img src="images/circle-unit-radians.svg" alt="Unit Circle Radians" height="397" width="496"></p>
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<div class="example">
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<h3>Example: What is sin(7<span class="times">π</span>/6) ?</h3>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/unit-circle-7pi-6.svg" alt="unit circle 7pi/6" height="158" width="182"></p>
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<p> </p>
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<p>Think "7<span class="times">π</span>/6 = <span class="times">π + </span><span class="times">π</span>/6", then make a sketch.</p>
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<p>We can then see it is <b>negative</b> and is the "short" value: −½</p>
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</div>
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<p> </p>
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<div class="questions">7708, 7709, 7710, 7711, 8903, 8904, 8906, 8907, 8905, 8908</div>
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<p> </p>
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<div class="fun">
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<h3>Footnote: where do the values come from?</h3>
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<p>We can use the equation <span class="large">x<sup>2</sup> + y<sup>2</sup> = 1</span> to find the lengths of <b>x</b> and <b>y</b> (which are equal to <b>cos</b> and <b>sin</b> when the radius is <b>1</b>):</p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/triangle-45.gif" alt="triangle 45 inside unit circle" height="225" width="232"></p>
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<h3>45 Degrees</h3>
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<p>For 45 degrees, x and y are equal, so <b>y=x</b>:</p>
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<div class="so">x<sup>2</sup> + x<sup>2</sup> = 1</div>
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<div class="so">2x<sup>2</sup> = 1</div>
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<div class="so"> x<sup>2</sup> = ½</div>
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<div class="so"> x = y = √(½)</div>
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<div style="clear:both"></div>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/triangle-30-60.gif" alt="triangle 30 60 inside unit circle" height="244" width="233"></p>
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<h3>60 Degrees</h3>
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<p>Take an <a href="../triangle.html">equilateral triangle</a> <i>(all sides are equal and all angles are 60°)</i> and split it down the middle.</p>
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<p>The "x" side is now <b>½</b>,</p>
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<p>And the "y" side is:</p>
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<div class="so">(½)<sup>2</sup> + y<sup>2</sup> = 1</div>
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<div class="so">¼ + y<sup>2</sup> = 1</div>
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<div class="so">y<sup>2</sup> = 1-¼ = ¾ </div>
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<div class="so">y = √(¾)</div>
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<h3>30 Degrees</h3>
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<p>30<i>°</i> is just 60<i>°</i> with x and y swapped, so <b>x = √(¾)</b> and <b>y = ½</b></p>
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<p>And:</p>
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<div class="center large"><span style="font-size:120%;">√</span><span class="overline">1/2</span> =
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<span style="font-size:120%;">√</span><span class="overline">2/4</span> = <span class="intbl"><em><span style="font-size:120%;">√</span><span class="overline">2</span></em><strong><span style="font-size:120%;">√</span><span class="overline">4</span></strong></span> = <span class="intbl"><em><span style="font-size:120%;">√</span><span class="overline">2</span></em><strong>2</strong></span></div>
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<p>Also:</p>
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<div class="center large"><span style="font-size:120%;">√</span><span class="overline">3/4</span> =
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<span class="intbl"><em><span style="font-size:120%;">√</span><span class="overline">3</span></em><strong><span style="font-size:120%;">√</span><span class="overline">4</span></strong></span> = <span class="intbl"><em><span style="font-size:120%;">√</span><span class="overline">3</span></em><strong>2</strong></span></div>
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<p>And here is the result (same as before):</p>
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<div class="beach">
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<table align="center" cellspacing="0" cellpadding="5" border="1">
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<tbody>
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<tr style="text-align:center;">
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<th width="100">Angle</th>
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<th width="100">Cos</th>
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<th width="100">Sin</th>
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<th width="100"><i>Tan=Sin/Cos</i></th>
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</tr>
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<tr style="text-align:center;">
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<th width="100">30<i>°</i></th>
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<td style="width:100px;"><span class="intbl"><em>√3</em><strong>2</strong></span></td>
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<td><span class="intbl"><em>1</em><strong>2</strong></span></td>
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<td style="width:100px;"><span class="intbl">
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|
<em>1</em>
|
|
<strong>√3</strong>
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|
</span>
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|
= <span class="intbl">
|
|
<em>√3</em>
|
|
<strong>3</strong>
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</span></td>
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</tr>
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<tr style="text-align:center;">
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<th width="100">45<i>°</i></th>
|
|
<td style="width:100px;"><span class="intbl"><em>√2</em><strong>2</strong></span></td>
|
|
<td><span class="intbl"><em>√2</em><strong>2</strong></span></td>
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<td style="width:100px;">1</td>
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</tr>
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<tr style="text-align:center;">
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<th width="100">60<i>°</i></th>
|
|
<td style="width:100px;"><span class="intbl"><em>1</em><strong>2</strong></span></td>
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|
<td><span class="intbl"><em>√3</em><strong>2</strong></span></td>
|
|
<td style="width:100px;">√3</td>
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</tr>
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</tbody></table><br>
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</div><p></p>
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</div>
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<p> </p>
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<div class="related">
|
|
<a href="circle.html">Circle</a> <span class="large">
|
|
<a href="../algebra/trig-interactive-unit-circle.html">Interactive Unit Circle</a></span>
|
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<a href="../algebra/trig-four-quadrants.html">Sine, Cosine and Tangent in Four Quadrants</a>
|
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<a href="../algebra/trigonometry-index.html">Trigonometry Index</a>
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</div>
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