QuestionUse a compound angle formula to determine a trigonometric expression that is equivalent to $\sin \left(x-\frac{4 \pi}{3}\right)$.

Studdy Solution
Simplify the expression using known trigonometric values for $\frac{4\pi}{3}$.
The angle $\frac{4\pi}{3}$ is in the third quadrant where sine is negative and cosine is negative. The reference angle is $\frac{\pi}{3}$.
$\cos \left(\frac{4 \pi}{3}\right) = -\cos \left(\frac{\pi}{3}\right) = -\frac{1}{2}$
$\sin \left(\frac{4 \pi}{3}\right) = -\sin \left(\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2}$
Substitute these values back into the expression:
$\sin \left(x-\frac{4 \pi}{3}\right) = \sin x \left(-\frac{1}{2}\right) - \cos x \left(-\frac{\sqrt{3}}{2}\right)$
Simplify:
$\sin \left(x-\frac{4 \pi}{3}\right) = -\frac{1}{2} \sin x + \frac{\sqrt{3}}{2} \cos x$
The trigonometric expression equivalent to $\sin \left(x-\frac{4 \pi}{3}\right)$ is $-\frac{1}{2} \sin x + \frac{\sqrt{3}}{2} \cos x$.