## <Success>Trigonometric functions: lecture 1</Success> <div style='float: left; width:50%;'> </div>
### <Success>Recall: basics of trigonometry</Success> <div style='float: left;text-align:left; width:60%;font-size:30px;'> <p><strong>Trigonometry</strong></p> <ul> <li>Trigono = Triangle</li> <li>Metry = Measure</li> </ul> <p>$\quad$</p> <ul> <li> <p>$ \text{Cosine} \to \cos \theta = \frac{\overline{AB}}{\overline{AC}}$</p> </li> <li> <p>$ \text{Sine} \to \sin \theta = \frac{\overline{BC}}{\overline{AC}}$</p> </li> <li> <p>$ \tan \theta = \frac{\overline{BC}}{\overline{AB}}$</p> </li> </ul> </div> <div style='float: right; width:39%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/e47e8554-6576-4650-8064-33e2869ef300/public"></p> </div> <div style='float: right; width:1%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/983a78f3-5294-410d-3de4-810876c14400/public"></p> </div>
### <Success>Heights & distances</Success> <div style='float: left;text-align:left; width:60%;font-size:30px;'> <ul> <li> <p>$ \tan 30^{\circ}= \frac{{H}} {{S+10}} = \frac{1}{\sqrt3} $</p> </li> <li> <p>$ \tan 60^{\circ}= \frac{{H}} {{S}} =\sqrt3 $</p> </li> </ul> </div> <div style='float: right; width:39%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/d711c141-e081-4375-0eb6-e25b3a21c600/public"></p> </div> <div style='float: right; width:1%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/f40fb906-15cb-451c-1460-37ef1b8a7700/public"></p> </div>
### <Success>Sides corresponding to angle θ </Success> <div style='float: left;text-align:left; width:60%;font-size:30px;'> <ul> <li>$AC = \text{Hypotenuse}$</li> <li>$BC = \text{Adjacent side}$</li> <li>$AB = \text{Opposite side}$</li> </ul> </div> <div style='float: left; width:49%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/d04e380b-493c-4b08-abcd-47e89f276800/public"></p> </div> <div style='float: right; width:1%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/2e45b32f-af93-43ae-f337-b0916a225700/public"></p> </div>
### <Success>Measure of angles</Success> <div style='float: left; width:60%;font-size:30px;'> <p>Anti-clockwise rotation $\Rightarrow$ positive angle</p> <p>Clockwise rotation $\Rightarrow$ negative angle</p> <p>There are two popular ways to measure angle:</p> <p>1)In Degree</p> <p>2)In Radian</p> </div> <div style='float: right; width:39%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/bd82e65b-2f16-451f-ec7f-9ca5cad77e00/public"></p> </div> <div style='float: right; width:1%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/ecea7891-8ac8-49de-20b0-23eabca92900/public"></p> </div>
### <Success>One degree</Success> <div style='float: left; width:50%;font-size:30px;'> <ul> <li>$1$ complete revolution $= 360^{\circ}$</li> <li>$1^{\circ} = (\frac{1}{360})^{\text{th}}$ of a complete revolution.</li> </ul> </div> <div style='float: right; width:50%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/69ad4fce-8b25-4929-af5e-59bdcf10c400/public"></p> </div> <div style='float: right; width:1%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/edbaaf62-215f-4992-3f6f-eaed7480b300/public"></p> </div>
### <Success>90° Rotation</Success> <div style='float: left; width:50%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/486bcc63-b142-4560-7f33-b19a9b48a400/public"></p> </div> <div style='float: right; width:01%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/c8c10b81-6d59-4434-cbd6-f86325d85800/public"></p> </div>
### <Success>Radian measure</Success> <div style='float: left; width:50%;font-size:30px;'> <p>Angle subtended at the centre by an arc of length 1 unit in a unit circle (circle of radius 1 unit) is said to have a measure of 1 radian.</p> </div> <div style='float: right; width:25%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/1562b3bb-0ccb-4cb7-2a54-a1c83e019300/public"></p> </div> <div style='float: right; width:25%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/ee3d2e1d-16b8-48c3-0522-4bb95aba3400/public"></p> </div> <div style='float: right; width:1%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/f57c052e-2be3-4eee-73d2-f39e74786d00/public"></p> </div>
##### <Success>Arc length and corresponding angle at centre</Success> <div style='float: left; width:50%;font-size:30px;'> <table> <thead> <tr> <th>Arc length</th> <th>Angle subtended at centre</th> </tr> </thead> <tbody> <tr> <td>1 UNIT</td> <td>1 RAD</td> </tr> <tr> <td>2 UNIT</td> <td>2 RAD</td> </tr> <tr> <td>3.17 UNIT</td> <td>3.17 RAD</td> </tr> <tr> <td>2 π UNIT</td> <td>2 π RAD</td> </tr> </tbody> </table> <p>1 complete Revolution = 2π RAD</p> </div> <div style='float: right; width:50%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/9233f730-f111-49b3-997a-3e95decc0e00/public"></p> </div> <div style='float: right; width:1%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/fe89f449-8fd1-4759-dac0-86a283dcf900/public"></p> </div>
##### <Success>Arc length and corresponding angle at centre</Success> <div style='float: left; width:50%;font-size:30px;'> <ul> <li>Radius of Inner Circle - 1 unit</li> <li>Radius of Outer Circle - R units.<br> $\quad$</li> </ul> <table> <thead> <tr> <th>Outer circle arc length</th> <th>angle subtended at centre</th> </tr> </thead> <tbody> <tr> <td>$x$ UNIT</td> <td>1 RAD</td> </tr> <tr> <td>2 π $x$</td> <td>2 π RAD</td> </tr> </tbody> </table> <p>$\Rightarrow$ 2 π R = 2 π $x$</p> <p>$\Rightarrow$ $x$ = R</p> </div> <div style='float: right; width:50%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/108db176-e99e-46c2-67ff-3841f1739500/public"></p> </div> <div style='float: right; width:1%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/9f4b3d77-295a-43b3-8ef0-5ff42786ee00/public"></p> </div>
##### <Success>Angle and arc length</Success> <div style='float: left;text-align:left;width:50%;font-size:30px;'> <table> <thead> <tr> <th>Angle</th> <th>Arc length</th> </tr> </thead> <tbody> <tr> <td>1 RAD</td> <td>R</td> </tr> <tr> <td>2 RAD</td> <td>2 R</td> </tr> <tr> <td>3.98 RAD</td> <td>3.98 R</td> </tr> <tr> <td>θ RAD</td> <td>θ R</td> </tr> </tbody> </table> <p>$\quad$</p> <ul> <li>l = θ R</li> </ul> </div> <div style='float: right; width:50%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/812afa40-bff9-47e6-0513-fa5a6dc03c00/public"></p> </div> <div style='float: right; width:1%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/fe04bdf5-c292-45f2-f126-8543cfc6ca00/public"></p> </div>
##### <Success>Relation between degree and radian</Success> <div style='float: left;text-align:left;width:100%;font-size:30px;'> <ul> <li>1 Complete revolution = 360° = 2 $\pi$ RAD<br> $ \therefore$ 1 RAD = $\frac{{360°}} {{2 \pi}}$ = 57.27° ($\pi$ = $\frac{22}{7}$)</li> <li>360° = 2$\pi$ RAD<br> $\Rightarrow \ 1^{\circ} = \frac{{2 \pi}} {{360}}$ RAD</li> </ul> </div> <p>======</p> <h3 id="successexamplesuccess"><Success>Example</Success></h3> <div style='float: left;text-align:left;width:100%;font-size:30px;'> <ol> <li>Find the value of $\frac{{\pi}} {{4}}$ RAD .</li> </ol> <p>$ \frac{{\pi}} {{4}}$ RAD = $\frac{{\pi}} {{4}} \times \frac{{360}^{\circ}} {{2\pi}}= 45°$</p> <ol start="2"> <li>Find the value of $ 135°.$</li> </ol> <p>$ 135° = 135° \times \frac{{2\pi}} {{360}}$ RAD</p> <p>$ = \biggl(\frac{{3\pi }} {{4}}\biggl) $RAD</p> </div>
### <Success>Example</Success> <div style='float: left; text-align:left;width:70%;font-size:30px;'> <p>The minute hand of a watch is 5 cm long. How far does its tip move in 42 minutes?</p> <p><strong>Solution:</strong></p> <p>1 complete revolution $=$ 2π RAD</p> <p>$\frac{42}{60}$ of 1 revolution</p> <p>$= \frac {42}{60} {2 \pi}$ RAD = ${1.4}{\pi}$ RAD</p> <p>We know that, l $=$ θ R</p> <p>l $= {1.4}{\pi} \ 5 \text{cm} \\ \ =22 \ \text{cm} $</p> </div> <div style='float: right; width:30%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/0fc2d142-9591-4f9f-b933-da9df60b5600/public"></p> </div> <div style='float: right; width:1%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/798dad95-3351-4e8f-f417-76307e4eb400/public"></p> </div>
### <Success>Domain and range of sine \& cosine</Success> <div style='float: left;text-align:left;width:50%;font-size:30px;'> <ul> <li>$ \sin (x) = b$</li> <li>$\cos (x) = a$</li> <li>$x \in \R, \ $ <strong>Domain $= \R$</strong></li> <li>Range of Sine: $-1 ≤ a ≤ 1$</li> <li>Range of Cosine $-1 ≤ b ≤ 1$</li> </ul> </div> <div style='float: right; width:50%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/b06b8a55-286d-4eee-9a6f-7733a7a40300/public"></p> </div> <div style='float: right; width:1%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/3f03828b-0b94-4322-1966-1e582d332400/public"></p> </div>
##### <Success>Property of sine & cosine function </Success> <div style='float: left;text-align:left;width:60%;font-size:30px;'> <p>$\sin \ x = {b} \ $ and $\ \cos \ x = {a} $</p> <p>$OA^2 + AP^2 = OP^2 $</p> <p>$a^2+b^2=1$</p> <p>$\cos^2x + \sin^2x = 1 $</p> </div> <div style='float: right; width:30%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/1a732efe-1d8d-429a-f28c-071525e01400/public"></p> </div> <p>======</p> <h4 id="successwhen-sin-x--0success"><Success>When sin x = 0?</Success></h4> <div style='float: left;text-align:left;width:90%;font-size:30px;'> $ \sin x = 0, x = ?$ <p>$ x=0, x= π $</p> <p>$ \sin(x) = \sin(x+k.2π)$</p> <p>for any integer k.</p> <p>$ \sin x = 0 \Leftrightarrow x = kπ$</p> </div> <div style='float: right; width:1%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/81094d19-b906-46fa-a9cf-572b3e670500/public"></p> </div>
<div> <h2><Success>Thank you</Success></h2> </div>