# Need help with finding exact value

• Mar 22nd 2010, 07:26 AM
perryman
Need help with finding exact value
Given that q = $\displaystyle sin – 1(– 3/5)$find the exact value of $\displaystyle sin (1/3π+ q).$
• Mar 22nd 2010, 08:22 AM
Soroban
Hello, perryman!

Quote:

Given that: .$\displaystyle q \,=\, \sin^{– 1}\left(-\,\frac{3}{5}\right)$

find the exact value of: .$\displaystyle \sin\left(q + \frac{\pi}{3}\right)$

We have: .$\displaystyle q \;=\;\sin^{-1}\left(-\,\frac{3}{5}\right)$

Then: . $\displaystyle \sin q \;=\;-\frac{3}{5} \;=\;\frac{opp}{hyp}$

$\displaystyle q$ is an angle in a right triangle with: $\displaystyle opp = -3,\;hyp = 5$
. . Pythagorus says: .$\displaystyle adj = \pm4$
Hence: .$\displaystyle \cos q \,=\,\pm\frac{4}{5}$

We have: .$\displaystyle \sin\left(q + \frac{\pi}{3}\right) \;=\;\sin q\cos\frac{\pi}{3} + \cos q\sin\frac{\pi}{3}$

. . . . . . . . . . . . . $\displaystyle =\;\left(-\frac{3}{5}\right)\left(\frac{1}{2}\right) + \left(\pm\frac{4}{5}\right)\left(\frac{\sqrt{3}}{2 }\right)$

. . . . . . . . . . . . . $\displaystyle =\;-\frac{3}{10} \pm\frac{4\sqrt{3}}{10}$

. . . . . . . . . . . . . $\displaystyle =\; \frac{-3 \pm 4\sqrt{3}}{10}$