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Chapter 5: Problem 36

Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. $$7 \pi / 12$$

Short Answer

Expert verified
The sine value for \(7\pi/12\) is \((\sqrt{6} - \sqrt{2})/4\), the cosine value for \(7\pi/12\) is \(-(\sqrt{6} + \sqrt{2})/4\), and the tangent value for \(7\pi/12\) is -1.

Step by step solution

01

Decompose the angle

Since \(7\pi/12\) cannot be simply represented with \(\pi/4\) or \(\pi/6\) units, we would have to decompose it into two angles for which we have known Trigonometric values. \(7\pi/12 = 3\pi/4 + \pi/6\)
02

Find the sine of the angle

Use the addition formula for sine, \(sin(a + b) = sin(a)cos(b) + cos(a)sin(b)\). Thus, \(sin(7\pi/12) = sin(3\pi/4)cos(\pi/6) + cos(3\pi/4)sin(\pi/6) = \(\sqrt{2}/2 * \sqrt{3}/2 + -\sqrt{2}/2 * 1/2 = (\sqrt{6} - \sqrt{2})/4\).
03

Find the cosine of the angle

Use the addition formula for cosine, \(cos(a + b) = cos(a)cos(b) - sin(a)sin(b)\). Thus, \(cos(7\pi/12) = cos(3\pi/4)cos(\pi/6) - sin(3\pi/4)sin(\pi/6) = -\sqrt{2}/2 * \sqrt{3}/2 - \sqrt{2}/2 * 1/2 = -\(\sqrt{6} + \sqrt{2})/4\).
04

Find the tangent of the angle

The tangent of an angle is the sine value divided by the cosine value, \(tan(x) = sin(x)/cos(x)\). Thus, \(tan(7\pi/12) = sin(7\pi/12) / cos(7\pi/12) = (\(\sqrt{6} - \sqrt{2})/4) / (-\sqrt{6} + \sqrt{2})/4 = -1\).

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