# Need help with finding exact value

• Mar 22nd 2010, 08:26 AM
perryman
Need help with finding exact value
Given that q = $sin – 1(– 3/5)$find the exact value of $sin (1/3π+ q)." alt="sin (1/3π+ q)." />
• Mar 22nd 2010, 09:22 AM
Soroban
Hello, perryman!

Quote:

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

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

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

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

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

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

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

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

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