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Chapter 6: Problem 22

Find the exact values of the sine, cosine, and tangent of the angle. $$-165^{\circ}$$

Short Answer

Expert verified
Hence, the exact values of the sine, cosine, and tangent of the angle -165掳 are \(-\sin(15^{\circ}) = - \frac{\sqrt{6} - \sqrt{2}}{4}\), \(-\cos(15^{\circ}) = - \frac{\sqrt{6} + \sqrt{2}}{4}\), and \(\tan(15^{\circ}) = 2 - \sqrt{3}\) respectively.

Step by step solution

01

Converting to Positive Angle

Convert -165 degrees into a positive angle by adding 360 degrees. The equivalent positive angle is \( 360 - 165 = 195 \) degrees.
02

Trigonometric Ratios for Sine

To find the sine of 195 degrees, you should know that sine values in the third quadrant (where 195 lies) are negative. For angle 伪 in the third quadrant, the sine value is equal to the negative sine of its reference angle (180掳 - 伪). Hence, \(\sin(-165^{\circ}) = -\sin(15^{\circ})\).
03

Trigonometric Ratios for Cosine

The cosine of angle 伪 in the third quadrant is also negative. Hence, \(\cos(-165^{\circ}) = -\cos(15^{\circ})\).
04

Trigonometric Ratios for Tangent

For the tangent, the ratio of sine/cosine gives the value of tangent. By substitution we get \( \tan(-165^{\circ}) = \frac{-\sin(15^{\circ})}{-\cos(15^{\circ})} = \tan(15^{\circ})\) as tangent is positive in the third quadrant.
05

Calculating the Values

The final step is substituting the known values of the trigonometric ratios at 15 degrees. Here, we will use the known values: \(\sin(15^{\circ}) = \frac{\sqrt{6} - \sqrt{2}}{4}\), \(\cos(15^{\circ}) = \frac{\sqrt{6} + \sqrt{2}}{4}\), and \(\tan(15^{\circ}) = 2 - \sqrt{3}\).

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