# Compound Angles

1. Nov 1, 2012

### odolwa99

In this question, my answer to part (a) is correct, & leads into (b). With (b) my answer is also correct, except for the sign. Can anyone help me figure what I need to do?

Many thanks.

1. The problem statement, all variables and given/known data

Q. $90^{\circ}<A<180^{\circ}$ such that $\sin(A+\frac{\pi}{6})+\sin(A-\frac{\pi}{6})=\frac{4\sqrt{3}}{5}$. Find (a) $\sin A$ & (b) $\tan A$

2. Relevant equations

3. The attempt at a solution

(a) $\sin A\cos \frac{\pi}{6}+\cos A\sin{\pi}{6}+\sin A\cos\frac{\pi}{6}-\cos A\sin\frac{\pi}{6}=\frac{4\sqrt{3}}{5}$
$2\sin A\cos\frac{\pi}{6}=\frac{4\sqrt{3}}{5}$
$\frac{2\sqrt{3}\sin A}{2}=\frac{4\sqrt{3}}{5}$
$2\sqrt{3}\sin A=\frac{8\sqrt{3}}{10\sqrt{3}}$
$\sin A=\frac{4}{5}$

(b) $\tan A=\frac{\sin A}{\cos A}$
$\sin^2A+cos^2A=1$
$(\frac{4}{5})^2+cos^2A=1$
$\cos^2A=1-\frac{16}{25}$
$\cos A=\sqrt{\frac{9}{25}}$
$\cos A=\frac{3}{5}$
$\tan A=\frac{4}{5}/\frac{3}{5}$
$\tan A=\frac{4}{3}$

Answer: (From text book): (b) $\frac{-4}{3}$

2. Nov 1, 2012

### CAF123

Use the condition given in the question: π/2 ≤ A ≤ π. if you like, draw the CAST diagram or sketch the cosine graph - what is the sign of cos here?
Note also $\sqrt{9/25} = +/- 3/5$

3. Nov 1, 2012

### odolwa99

Great. Thank you.