Question

Given cos 60° 1/2 use the trigonometric identities to find the exact value of each of the following. (a) sin 30° (b) sin²60° (c) sec π/3 (d) csc π/6

Answer 1

Using trigonometric identities and known values for special angles, we can find the exact values of sin 30°, sin² 60°, sec π/3, and csc π/6 based on the fact that cos 60° is 1/2.

Step-by-step explanation:

To find the exact values of the trigonometric functions given that cos 60° is 1/2, we use known trigonometric identities and the fact that the trigonometric functions for special angles such as 30°, 60°, and their radian equivalents are known.

• (a) sin 30° is √(1-cos²(60°)), which equals √(1-(1/2)²) = √(3/4) = 1/√2 or √2/2.
• (b) sin² 60° is 1-cos²(60°), which equals 1-(1/2)² = 3/4.
• (c) sec π/3 is reciprocal of cos(π/3), which is 2 because cos(π/3) = cos(60°) = 1/2.
• (d) csc π/6 is reciprocal of sin(π/6), which is 2 because sin(π/6) = sin(30°) = 1/2.