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Chapter 5: Problem 18

Find the exact values of the sine, cosine, and tangent of the angle. $$165^{\circ}=135^{\circ}+30^{\circ}$$

Short Answer

Expert verified
The exact value of \(\sin 165^\circ = -\frac{\sqrt{6} - \sqrt{2}}{4}\), \(\cos 165^\circ = -\frac{\sqrt{6} + \sqrt{2}}{4}\) and \(\tan 165^\circ = 2 - \sqrt{3}\).

Step by step solution

01

Break Down the Angles

Recognize the given angle 165 degrees as the sum of two known angles 135 degrees and 30 degrees.
02

Apply Sum Identities for Sine and Cosine

Using the trigonometric identities for the sum of two angles: \(\sin(胃_1+胃_2)= \sin(胃_1)\cos(胃_2)+\cos(胃_1)\sin(胃_2)\) and \(\cos(胃_1+胃_2)= \cos(胃_1)\cos(胃_2)-\sin(胃_1)\sin(胃_2)\), these identities can be applied to find the sine and cosine of 165 degrees.
03

Find the Values of Sine and Cosine for the Angles

The values of \(\sin(135^\circ)\), \(\cos(135^\circ)\), \(\sin(30^\circ)\) and \(\cos(30^\circ)\) are known and these need to be substituted into the formulae.
04

Substitute and Simplify to Find Sine and Cosine

Replace each value in the right side of both formulae and simplify to find the exact values of sine and cosine for 165 degrees.
05

Apply the Tan Identity & Simplify for the Tangent

Once the sine and cosine values are found, use the identity \(\tan(胃)= \frac{\sin(胃)}{\cos(胃)}\) to find the tangent of the angle, by division of sine 165 degrees over cosine 165 degrees. Simplify it to find the exact value for tan 165 degrees.

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