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Chapter 5: Problem 22

Find the exact values of the sine, cosine, and tangent of the angle. $$\frac{5 \pi}{12}$$

Short Answer

Expert verified
The sine of \(\frac{5\pi}{12}\) is \(\frac{\sqrt{6} + \sqrt{2}}{4}\), the cosine of \(\frac{5\pi}{12}\) is \(\frac{\sqrt{6} - \sqrt{2}}{4}\), and the tangent of \(\frac{5\pi}{12}\) is \(\frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}}\).

Step by step solution

01

Break-down the angle

We choose to break down the angle \(\frac{5\pi}{12}\) into a sum of two standard angles such that \(\frac{\pi}{4} + \(\frac{\pi}{6} = \frac{5\pi}{12}\).
02

Use trigonometric identity for sine

Apply the sum of angle formula for sine, which is: \(sin(A + B) = sinAcosB + cosAsinB\). Substitute A with \(\frac{\pi}{4}\) and B with \(\frac{\pi}{6}\) to get \(sin \frac{5\pi}{12} = sin(\frac{\pi}{4}) cos(\frac{\pi}{6}) + cos( \frac{\pi}{4}) sin(\frac{\pi}{6}) = \frac{\sqrt{6} + \sqrt{2}}{4}\).
03

Use trigonometric identity for cosine

Apply the sum of angle formula for cosine, which is: \(cos(A + B) = cosAcosB - sinAsinB\). Substitute A with \(\frac{\pi}{4}\) and B with \(\frac{\pi}{6}\) to get \(cos \frac{5\pi}{12} = cos(\frac{\pi}{4}) cos(\frac{\pi}{6}) - sin(\frac{\pi}{4})sin(\frac{\pi}{6}) = \frac{\sqrt{6} - \sqrt{2}}{4}\).
04

Calculate the tangent

Finally, calculate the tangent of the angle using the formula: \(tan(x) = \frac{sin(x)}{cos(x)}\). Substitute \(x\) with \(\frac{5\pi}{12}\) to get \(tan\frac{5\pi}{12} = \frac{sin\frac{5\pi}{12}}{cos\frac{5\pi}{12}} = \frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}}\).

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