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Chapter 4: Problem 57

Evaluate the sine, cosine, and tangent of the angle without using a calculator. $$ -150^{\circ} $$

Short Answer

Expert verified
Sine of -150掳 is -0.5, Cosine of -150掳 is -\( \frac{\sqrt{3}}{2} \) and Tangent of -150掳 is \( -\frac{1}{\sqrt{3}} \).

Step by step solution

01

Conversion of negative angle to positive angle

By adding 360掳 to -150掳, it becomes 360掳-150掳=210掳. Thus -150掳 and 210掳 represents the same position in the unit circle.
02

Identify the quadrant and sign

210掳 lies in the third quadrant (180掳 to 270掳) where sine is negative, cosine is negative and tangent is positive.
03

Identify the reference angle

The reference angle for the given angle is the acute angle it makes with the x-axis. For 210掳, it will be 210掳 - 180掳 = 30掳.
04

Find sine, cosine, and tangent

Now, using the sine, cosine, and tangent values of 30掳 from the unit circle and apply the signs we found from the quadrant in which 210掳 lies. Sine of 30掳 is 0.5, Cosine of 30掳 is \( \frac{\sqrt{3}}{2} \), and the tangent of 30掳 is \( \frac{1}{\sqrt{3}} \). Therefore, sine of -150掳 or 210掳 is -0.5, cosine is -\( \frac{\sqrt{3}}{2} \) and tangent is \( -\frac{1}{\sqrt{3}} \).

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