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

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

Options:

Correct Answer: √3/2

Solution:

If x = cos^(-1)(1/2), then cos(x) = 1/2, which corresponds to x = π/3. Therefore, sin(x) = sin(π/3) = √3/2.

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

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

Step 1: Understand that cos^(-1)(1/2) means we are looking for an angle x where the cosine of x is 1/2.
Step 2: Recall the unit circle or the values of cosine for common angles. We know that cos(π/3) = 1/2.
Step 3: Therefore, we can conclude that x = π/3.
Step 4: Now, we need to find sin(x). Since we found x = π/3, we need to calculate sin(π/3).
Step 5: Recall the value of sin(π/3). It is known that sin(π/3) = √3/2.
Step 6: Thus, the value of sin(x) is √3/2.

Inverse Trigonometric Functions – Understanding how to interpret and evaluate inverse trigonometric functions, specifically cos^(-1) and its relationship to angles and their sine values.
Trigonometric Identities – Applying the relationship between sine and cosine functions, particularly using the identity sin^2(x) + cos^2(x) = 1.