Evaluate sin(cos^(-1)(1/2)).

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Question: Evaluate sin(cos^(-1)(1/2)).

Options:

Correct Answer: √3/2

Solution:

sin(cos^(-1)(1/2)) = √(1 - (1/2)^2) = √(3/4) = √3/2.

Evaluate sin(cos^(-1)(1/2)).

Evaluate sin(cos^(-1)(1/2)).

Correct Answer: √3/2

Step 1: Understand that cos^(-1)(1/2) means we are looking for an angle whose cosine is 1/2.
Step 2: Recall that the angle whose cosine is 1/2 is 60 degrees or π/3 radians.
Step 3: We need to find the sine of this angle, sin(π/3).
Step 4: Use the Pythagorean identity: sin(θ) = √(1 - cos^2(θ)).
Step 5: Substitute cos(θ) = 1/2 into the identity: sin(θ) = √(1 - (1/2)^2).
Step 6: Calculate (1/2)^2 = 1/4.
Step 7: Now substitute back: sin(θ) = √(1 - 1/4).
Step 8: Simplify: 1 - 1/4 = 3/4.
Step 9: So, sin(θ) = √(3/4).
Step 10: Finally, simplify √(3/4) to get √3/2.

Inverse Trigonometric Functions – Understanding how to evaluate the sine of an angle given its cosine value using the inverse cosine function.
Pythagorean Identity – Using the identity sin²(θ) + cos²(θ) = 1 to find the sine value from the cosine value.