Question

If 3sin$\beta$ = sin(2$\alpha$ + $\beta$), then tan ($\alpha$ + $\beta$) is equal to

• A. 2tan
• B. 2tan
• C. tan + tan
• D. None of these

Answer 1

The correct option is B

2tan

We will try to solve this in two ways. First one slightly tricky compared to the second.

We want to find tan ($\alpha$ + $\beta$).

If we try to expand tan($\alpha$ + $\beta$), we don't know the values of tan $\alpha$ and tan $\beta$. We will re-write the expression as

3 sin ($\alpha$ + $\beta$ - $\alpha$)= sin ($\alpha$ + $\alpha$ + $\beta$)

Here, we are creating the angle ($\alpha$ + $\beta$) on both sides. Later we will divide by cos ($\alpha$ + $\beta$) on both sides to get tan ($\alpha$ + $\beta$).

$\Rightarrow$ 3(sin($\alpha$ + $\beta$) cos $\alpha$ - (cos + $\beta$) sin$\alpha$)

= sin$\alpha$ cos ($\alpha$ + $\beta$) +cos $\alpha$ sin ($\alpha$ + $\beta$)

Dividing by cos ($\alpha$ + $\beta$) through out to get tan ($\alpha$ + $\beta$),

$\Rightarrow$ 3 tan ($\alpha$ + $\beta$) cos $\alpha$ - 3sin$\alpha$ = sin $\alpha$+tan ($\alpha$ + $\beta$) cos$\alpha$

$\Rightarrow$ 2 cos $\alpha$ tan ($\alpha$ + $\beta$) = 4 sin $\alpha$

$\Rightarrow$ tan ($\alpha$ + $\beta$) = 2 tan $\alpha$

In this method, important step is rewriting the expression.

Method 2

Inthis method, we will apply componendo dividendo after taking the trigonometric ratios to one side.

$\frac{sin(2\alpha +\beta )}{sin\beta }$ = 3

(This is like a common step in problem like this.If you have done similar problems before, this step is kind of intutive)

$\frac{sin(2\alpha +\beta )+sin\beta }{sin(2\alpha +\beta )-sin\beta }$ = $\frac{3+1}{3-1}$

$\frac{2sin(\alpha +\beta )cos\alpha }{2sin\alpha \times cos(\alpha +\beta )}$ = 2

$\Rightarrow$ tan($\alpha$ + $\beta$) = 2 tan$\alpha$

key steps/concepts: (1) Componendo dividend or

(2) $\beta$ = ($\alpha$ + $\beta$) - $\alpha$ and 2$\alpha$ + $\beta$ = ($\alpha$ + $\beta$) + $\alpha$