Problem 57 - Precalculus: A Concise Course

Question

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

Step-by-Step Solution

Answer

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

Step 1: Conversion of negative angle to positive angle

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

Step 2: Identify the quadrant and sign

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

Step 3: Identify the reference angle

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

Step 4: Find sine, cosine, and tangent

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