If x = cos^(-1)(1/2), then what is the value of sin^(-1)(√(1 - (1/2)^2))?

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Question: If x = cos^(-1)(1/2), then what is the value of sin^(-1)(√(1 - (1/2)^2))?

Options:

Correct Answer: π/3

Solution:

Since cos^(-1)(1/2) = π/3, we have sin^(-1)(√(1 - (1/2)^2)) = sin^(-1)(√(3/4)) = π/3.

If x = cos^(-1)(1/2), then what is the value of sin^(-1)(√(1 - (1/2)^2))?

If x = cos^(-1)(1/2), then what is the value of sin^(-1)(√(1 - (1/2)^2))?

Step 1: Understand that x = cos^(-1)(1/2) means we are looking for an angle whose cosine is 1/2.
Step 2: Recall that cos(π/3) = 1/2. Therefore, x = π/3.
Step 3: Now, we need to find sin^(-1)(√(1 - (1/2)^2)).
Step 4: Calculate (1/2)^2, which is 1/4.
Step 5: Subtract 1/4 from 1: 1 - 1/4 = 3/4.
Step 6: Now, take the square root of 3/4: √(3/4) = √3/2.
Step 7: We need to find sin^(-1)(√3/2).
Step 8: Recall that sin(π/3) = √3/2. Therefore, sin^(-1)(√3/2) = π/3.
Step 9: Conclude that sin^(-1)(√(1 - (1/2)^2)) = π/3.

Inverse Trigonometric Functions – Understanding the relationship between cosine and sine values through their inverse functions.
Pythagorean Identity – Using the identity sin^2(x) + cos^2(x) = 1 to derive values of sine from cosine.