Question

If (a-b) sin($\theta +\varphi$) = (a+b) sin($\theta -\varphi$),find tan$\theta$

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

We solved similar problems and saw that when we have ($\theta +\alpha$) as and ($\theta +\alpha$) as angles and (a-b) and (a+b) as coefficients, applying componendo - dividendo simplifies the expression (not always!)

$\Rightarrow$ $\frac{a+b}{a-b}$ = $\frac{sin(\theta +\varphi )}{sin(\theta -\varphi )}$

$\Rightarrow$ $\frac{a+b+a-b}{a+b-(a-b)}$ = $\frac{sin(\theta +\varphi )+sin(\theta +\varphi )}{sin(\theta -\varphi )-sin(\theta -\varphi )}$

$\Rightarrow$ $\frac{a}{b}$ = $\frac{2sin\theta cos\varphi }{2cos\theta sin\varphi }$

$\Rightarrow$ $\frac{a}{b}$ = $\frac{tan\theta }{tan\varphi }$

$\Rightarrow$ tan$\theta$ = $\frac{a}{b}$ tan$\varphi$