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Chapter 5: Problem 12

Find the exact values of the sine, cosine, and tangent of the angle. $$\frac{7 \pi}{12}=\frac{\pi}{3}+\frac{\pi}{4}$$

Short Answer

Expert verified
The exact values for sine, cosine, and tangent of the angle \( \frac{7\pi}{12}\) are: \(\sin(\frac{7\pi}{12}) = \sqrt{6}+\sqrt{2}\), \(\cos(\frac{7\pi}{12}) = \sqrt{2}-\sqrt{6}\), and \(\tan(\frac{7\pi}{12}) = -1\).

Step by step solution

01

Use the formula for sine of sum of angles

The formula for sine of sum of two angles is: \( \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) \).\n Use this formula for the given angle: \(\sin(\frac{\pi}{3}+\frac{\pi}{4})\)= \(\sin(\frac{\pi}{3})\cos(\frac{\pi}{4}) + \cos(\frac{\pi}{3})\sin(\frac{\pi}{4})\).
02

Substitute known sinus and cosinus values and simplify

Substitute the known values: \(\sin(\frac{\pi}{3})\)= \(\sqrt{3}/2\), \(\cos(\frac{\pi}{3})\)= \(1/2\), \(\sin(\frac{\pi}{4})\)= \(\cos(\frac{\pi}{4})\)= \(1/\sqrt{2}\).\n So the expression simplifies to: \(\sqrt{3}/2 * 1/\sqrt{2} + 1/2 * 1/\sqrt{2} = \sqrt{3}/2\sqrt{2} + 1/2\sqrt{2} = \sqrt{6} + \sqrt{2} / 4\).
03

Apply the cosine addition formula

The formula for cosine of sum two angles is: \( \cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b) \).\n Use this formula for the given angle: \(\cos(\frac{\pi}{3}+\frac{\pi}{4})\)= \(\cos(\frac{\pi}{3})\cos(\frac{\pi}{4}) - \sin(\frac{\pi}{3})\sin(\frac{\pi}{4})\).
04

Substitute known sinus and cosinus values and simplify

Substitute the known values: \(\cos(\frac{\pi}{3})\)= \(1/2\), \(\sin(\frac{\pi}{3})\)= \(\sqrt{3}/2 \).\n So the expression simplifies to: \(1/2 * 1/\sqrt{2} - \sqrt{3}/2 * 1/\sqrt{2} = \sqrt{2}-\sqrt{6} / 4\).
05

Use the quotient of sin(a+b)/cos(a+b) for tangent

The tangent of an angle is given by the quotient of sine over cosine. \(\tan(\frac{\pi}{3}+\frac{\pi}{4})\)= \(\sin(\frac{\pi}{3}+\frac{\pi}{4}) / \cos(\frac{\pi}{3}+\frac{\pi}{4})=\)(\(\sqrt{6} + \sqrt{2}\) / \(\sqrt{2}-\sqrt{6}\)). After rationalizing the denominator, \(\tan(\frac{\pi}{3}+\frac{\pi}{4})=\frac{\sqrt{6} + \sqrt{2}}{2}\).
06

Supply necessary simplifications

Under the radicals, simplify \(2\sqrt{6} + 2\sqrt{2}\) to \(\sqrt{6} + \sqrt{2}\) and \(2\sqrt{2} - 2\sqrt{6}\) to \(\sqrt{2} - \sqrt{6}\).
07

State final exact values

The exact values of the trigonometric functions of the angle \(\frac{7 \pi}{12}=\frac{\pi}{3}+\frac{\pi}{4}\) are then: \(\sin(\frac{7\pi}{12}) = \sqrt{6}+\sqrt{2}\), \(\cos(\frac{7\pi}{12}) = \sqrt{2}-\sqrt{6}\), and \(\tan(\frac{7\pi}{12}) = -1\).

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