### <Success>Trigonometric functions <br> lecture - 2</Success> <div style='float: left; width:50%;'> </div>

### <Success>$\text{General solution for} \ \cos x = 0$</Success> <div style='float: left; width:70%;font-size:30px;'> <p>$ \cos x = 0,$</p> <p>$ x = \frac{\pi}{2}, \frac{3\pi}{2}$</p> <p>$ \cos(x) = \cos\ (x+k. 2\pi) $</p> <p>$ x = \biggl ( n+ \frac{1}{2}\biggl) {\pi} $ ,n is integer.</p> </div> <div style='float: right; width:30%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/52f619a7-7d75-47d3-8683-73dcda9b6000/public"></p> </div> <div style='float: right; width:1%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/e318149c-a312-4e65-e070-0114bf765a00/public"></p> </div>

#### <Success>$\text{Sine and cosine of some angles}$</Success> <div style='float: left; width:50%;font-size:30px;'> <p>$ \cos {\theta} = \frac {AB}{AC} $</p> <p>$ \sin \biggl (\frac {\pi}{2}-{\theta} \biggl ) = \frac {AB}{AC} $</p> <p>$ \cos {\theta} = \sin \biggl (\frac {\pi}{2}-{\theta} \biggl ) $</p> <p>$ \cos \biggl (\frac {\pi}{4}\biggl ) = \frac {1} {\sqrt{2} }$</p> <p>$ \sin \biggl (\frac {\pi}{4} \biggl ) = \frac {1}{\sqrt{2} } $</p> </div> <div style='float: right; width:25%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/93a5a8df-d06d-4d2f-1a86-5c6f9f3ac000/public"></p> </div> <div style='float: right; width:25%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/404e53da-de72-420e-2641-fdf1043c7c00/public"></p> </div> <div style='float: right; width:1%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/eec51900-1948-4487-64bc-691b70283800/public"></p> </div>

#### <Success>$\text{Proving} \sin({\pi}/6) = 1/2$</Success> <div style='float: left; width:50%;font-size:30px;'> <p>$\triangle$ ABC and $\triangle$ ABD are congruent</p> <p>$\frac {\pi}{3} + {\theta}+{\theta}= {\pi},\ {\theta}= \frac {\pi} {3} $</p> <p>$\Rightarrow \triangle ADC$ is equilateral</p> <p>$\Rightarrow CD= 1$ unit</p> <p>$\sin \biggl (\frac {\pi}{6} \biggl )= \frac {1}{2} $</p> </div> <div style='float: right; width:50%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/4cd2d642-d5fa-416f-e25f-9de43ba55e00/public"></p> </div>

#### <Success>$\text{Relation between} \sin (x) \\& \sin (-x)$</Success> <div style='float: left; width:60%;font-size:30px;'> <p>$ \sin (x) = b$</p> <p>$ \sin (-x) = d$</p> <p>$\triangle OAP \cong \triangle OAQ$</p> <p>$ c = a $</p> <p>$ d = -b $</p> <p>$ \sin (-x) = -b = -\sin (x) $</p> <p>$\sin (-x) = -\sin(x) $</p> <p>$ f (-x) = -f(x) $ $ \Leftrightarrow$ Odd functions</p> </div> <div style='float: right; width:40%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/d0460a3d-f7d1-42f9-b9d2-29e0a1567600/public"></p> </div> ====== #### <Success>$\text{Relation between} \cos (x) \\& \cos (-x)$</Success> <div style='float: left; width:100%;font-size:30px;'> <p>$ \cos (x) = a $</p> <p>$ \cos (-x) = a $</p> <p>$ \cos (x) = \cos (-x) $</p> <p>$ f (x) = f (-x) $ $ \Leftrightarrow$ Even functions</p> </div> <div style='float: right; width:1%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/9c0b8d29-c43e-425c-463a-5bc401334500/public"></p> </div>

#### <Success>$\text{Range of sine and cosine in different}$ $\text{quadrants}$</Success> <div style='float: left;text-align:left; width:70%;font-size:30px;'> <table> <thead> <tr> <th>Quadrant</th> <th>$\sin(x)$</th> <th>$\cos(x)$</th> </tr> </thead> <tbody> <tr> <td>$ \quad \text{(I)}\\ 0 ≤ x< \frac{\pi}{2} $</td> <td>[0,1)</td> <td>(0,1]</td> </tr> <tr> <td>$ \quad \text{(II)}\\ \frac {\pi} {2} ≤ x <{\pi} $</td> <td>(0,1]</td> <td>(-1,0]</td> </tr> <tr> <td>$ \quad \text{(III)}\\ {\pi} ≤ x <\frac {3\pi} {2} $</td> <td></td> <td></td> </tr> <tr> <td>$ \quad \text{(IV)}\\ \frac {3\pi} {2} ≤ x < 2{\pi } $</td> <td></td> <td></td> </tr> </tbody> </table> </div> <div style='float: right; width:30%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/ffa95334-65f0-4a1f-6a70-0c3ab91cf500/public"></p> </div> <div style='float: right; width:1%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/aa07eb0a-7cf2-47f4-4755-f3306df27900/public"></p> </div>

### <Success>$\text{Graph of sin x}$</Success> <div style='float: left; width:50%;'> </div> <div style='float: right; width:90%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/85e6ff16-861d-4c58-0279-1683e577fa00/public"></p> </div> <div style='float: right; width:1%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/8e266ca0-bfe6-4c9e-83c8-ebe2f6541c00/public"></p> </div>

### <Success>$\text{Formula for} \cos (x-y) $</Success> <div style='float: left; width:50%;font-size:30px;'> <p>$ \triangle OPQ \cong \triangle OAR$</p> <p>$ OQ=OR $</p> <p>$ OP=OA $</p> <p>$ \angle POQ= \angle AOR $</p> <p>$ \angle POQ= x-y $</p> <p>$ QP ^2= AR ^2 $</p> </div> <div style='float: right; width:50%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/3aaac4d6-dcf7-4fb3-aac5-b03def8a0700/public"></p> </div> <div style='float: right; width:1%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/69a9a2a9-8e55-4ccf-2bba-db3a13ccdc00/public"></p> </div>

### <Success>$\text{Formula for} \cos (x-y) $</Success> <div style='float: left; width:100%;font-size:30px;'> <p>$ QP ^2= (\cos x - \cos y) ^2 + (\sin x - \sin y) ^2 $</p> <p>$ AR ^2= [\cos (x -y) - 1] ^2 + \sin ^2 (x-y) $</p> <p>As,$ QP ^2 = (\cos x - \cos y) ^2 + (\sin x - \sin y ) ^2$</p> <p>$= \cos ^2 x + \cos ^2 y -2 \cos x \cos y + \sin ^2 x + \sin ^2 y -2 \sin x \sin y $</p> <p>$ = 1 + 1 -2 \cos x \cos y -2 \sin x \sin y $</p> <div> ====== ### <Success>$\text{Formula for} \cos (x-y) $</Success> <div style='float: left; width:100%;font-size:30px;'> <p>$ AR ^2 = [\cos (x- y) -1] ^2 + \sin ^2 (x -y) $</p> <p>$ = \cos ^2 (x- y) +1 -2 \cos (x -y) + \sin ^2 (x-y)$</p> <p>$ = 1 + 1 -2 \cos (x -y ) $</p> <p>$\because QP ^2 = AR ^2$</p> <p>$\Rightarrow \cos (x- y) = \cos x \cos y + \sin x \sin y $</p> </div>

### <Success>$\text{Formula for} \cos (x+y) $</Success> <div style='float: left; width:100%;font-size:30px;'> <p>$\because \cos (x- y) = \cos x \cos y + \sin x \sin y $</p> <p>$ \cos (x+ y) = \cos (x - (- y)) $</p> <p>$ \cos (x+ y) = \cos x \cos (- y) + \sin (x) \sin (-y) $</p> <p>$ \cos (x+ y) = \cos x \cos y - \sin (x) \sin (y) $</p> </div>

<div> <h2><Success>Thank you</Success></h2> </div>