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Chapter 5: Problem 13

Find the exact values of the sine, cosine, and tangent of the angle. $$\frac{11 \pi}{12}=\frac{3 \pi}{4}+\frac{\pi}{6}$$

Short Answer

Expert verified
Thus, the exact values of the sine, cosine, and tangent of the angle \(\frac{11 \pi}{12}\) are \(\frac{\sqrt{6} - \sqrt{2}}{4}\), \(\frac{-\sqrt{6} - \sqrt{2}}{4}\), and \(-1\) respectively.

Step by step solution

01

- Rewrite the angle

We note that the angle \(\frac{11 \pi}{12}\) can be represented as the sum of the angles \(\frac{3 \pi}{4}\) and \(\frac{\pi}{6}\). So, we can write \(\frac{11 \pi}{12}=\frac{3 \pi}{4}+\frac{\pi}{6}\).
02

- Determine the sine of the angle

Use the sine addition formula, \(\sin(a + b) = \sin a \cos b + \cos a \sin b\), to obtain \(\sin(\frac{11 \pi}{12}) = \sin(\frac{3 \pi}{4}+\frac{\pi}{6}) = \sin(\frac{3 \pi}{4})\cos(\frac{\pi}{6}) + \cos(\frac{3 \pi}{4})\sin(\frac{\pi}{6}) = \frac{\sqrt{2}}{2} * \frac{\sqrt{3}}{2} + \frac{-\sqrt{2}}{2} * \frac{1}{2} = \frac{\sqrt{6} - \sqrt{2}}{4}\).
03

- Determine the cosine of the angle

Use the cosine addition formula, \(\cos(a + b) = \cos a \cos b - \sin a \sin b\), to obtain \(\cos(\frac{11 \pi}{12}) = \cos(\frac{3 \pi}{4}+\frac{\pi}{6}) = \cos(\frac{3 \pi}{4})\cos(\frac{\pi}{6}) - \sin(\frac{3 \pi}{4})\sin(\frac{\pi}{6}) = \frac{-\sqrt{2}}{2} * \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} * \frac{1}{2} = \frac{-\sqrt{6} - \sqrt{2}}{4}\).
04

- Determine the tangent of the angle

Use the tangent identity, \(\tan(a) = \frac{\sin a}{\cos a}\), to determine \(\tan(\frac{11 \pi}{12}) = \frac{\sin(\frac{11 \pi}{12})}{\cos(\frac{11 \pi}{12})} = \frac{\frac{\sqrt{6} - \sqrt{2}}{4}}{\frac{-\sqrt{6} - \sqrt{2}}{4}} = -1\).

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