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Chapter 5: Problem 9

Find the exact value of each expression. (a) \(\sin \left(\frac{7 \pi}{6}-\frac{\pi}{3}\right)\) (b) \(\sin \frac{7 \pi}{6}-\sin \frac{\pi}{3}\)

Short Answer

Expert verified
The exact value for (a) is \( \frac{1}{2} \), and the exact value for (b) is \( -1/2 - \sqrt{3}/2 \).

Step by step solution

01

Evaluate the expression inside the sine function for (a)

First, an angle subtraction should be performed inside the sine function. This gives: \( \frac{7 \pi}{6}-\frac{\pi}{3} = \frac{7\pi - 2\pi}{6} = \frac{5\pi}{6} \)
02

Find the sine value of the angle

Now use the unit circle or known values of sine at common angles to find that \( \sin \frac{5\pi}{6} = \frac{1}{2} \)
03

Determine the individual sine values for (b)

No operations inside the sine are needed here. Simply take the individual sine values. This gives \( \sin \frac{7\pi}{6} = -1/2 \) and \( \sin \frac{\pi}{3} = \sqrt{3}/2 \)
04

Subtract the two sine values

Finally, the two computed sine values must be subtracted from each other, giving \( -1/2 - \sqrt{3}/2 = -1/2 - \sqrt{3}/2 \)

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