10 3 Further Applications – Class 11TH Math

10.3 Further Applications of Basic Identities

Example 5

Prove $\sin(\alpha+\beta)\sin(\alpha-\beta) = \sin^2\alpha - \sin^2\beta = \cos^2\beta - \cos^2\alpha$ .

Expand as a difference of squares in $\sin\alpha\cos\beta$ and $\cos\alpha\sin\beta$ , then rewrite using $1 - \sin^2$ / $1 - \cos^2$ .

Example 6

All six trig functions of $105^\circ = 60^\circ + 45^\circ$ :

\sin 105^\circ = \dfrac{\sqrt{3}+1}{2\sqrt{2}},\quad \cos 105^\circ = \dfrac{1-\sqrt{3}}{2\sqrt{2}},\quad \tan 105^\circ = \dfrac{1+\sqrt{3}}{1-\sqrt{3}}

and reciprocals for $\cot$ , $\operatorname{cosec}$ , $\sec$ .

Example 7

$\dfrac{\cos 11^\circ + \sin 11^\circ}{\cos 11^\circ - \sin 11^\circ} = \tan 56^\circ$

(Write $\tan 56^\circ = \tan(45^\circ+11^\circ)$ and simplify.)

Example 8

Given $\cos\alpha = -\dfrac{7}{25}$ (Quad II) and $\tan\beta = \dfrac{12}{5}$ (Quad III):

$\sin\alpha = \dfrac{24}{25}$ , $\cos\beta = -\dfrac{5}{13}$ , $\sin\beta = -\dfrac{12}{13}$ .

\sin(\alpha+\beta) = -\dfrac{36}{325},\quad \cos(\alpha+\beta) = \dfrac{323}{325}

Terminal arm of $\alpha+\beta$ lies in Quad IV.

Example 9

In $\triangle ABC$ , $\alpha+\beta+\gamma = 180^\circ$ :

(i) $\tan\alpha + \tan\beta + \tan\gamma = \tan\alpha\tan\beta\tan\gamma$

(ii) $\tan\dfrac{\alpha}{2}\tan\dfrac{\beta}{2} + \tan\dfrac{\beta}{2}\tan\dfrac{\gamma}{2} + \tan\dfrac{\gamma}{2}\tan\dfrac{\alpha}{2} = 1$

Example 10

$3\sin\theta + 4\cos\theta = 5\sin(\theta + \phi)$ where $\tan\phi = \dfrac{4}{3}$ and $\phi$ is in Quad I ( $r = 5$ , $r\cos\phi = 3$ , $r\sin\phi = 4$ ).