If sin 2A = 2 sin A cos A, what is the value of sin 2A when sin A = 1/2?

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Question: If sin 2A = 2 sin A cos A, what is the value of sin 2A when sin A = 1/2?

Options:

Correct Answer: 1

Solution:

Using the double angle formula, sin 2A = 2(1/2)(√(1 - (1/2)²)) = 2(1/2)(√(3/4)) = 1.

If sin 2A = 2 sin A cos A, what is the value of sin 2A when sin A = 1/2?

If sin 2A = 2 sin A cos A, what is the value of sin 2A when sin A = 1/2?

Step 1: Identify the given value. We know that sin A = 1/2.
Step 2: Use the double angle formula for sine, which is sin 2A = 2 sin A cos A.
Step 3: Substitute sin A into the formula. Since sin A = 1/2, we have sin 2A = 2(1/2) cos A.
Step 4: Calculate cos A. We know that sin² A + cos² A = 1. So, (1/2)² + cos² A = 1.
Step 5: Simplify the equation. (1/4) + cos² A = 1, which means cos² A = 1 - 1/4 = 3/4.
Step 6: Take the square root to find cos A. cos A = √(3/4) = √3/2.
Step 7: Substitute cos A back into the double angle formula. Now we have sin 2A = 2(1/2)(√3/2).
Step 8: Simplify the expression. sin 2A = 1 * (√3/2) = √3/2.

Double Angle Formula – The question tests the understanding of the double angle formula for sine, which states that sin 2A = 2 sin A cos A.
Trigonometric Values – It assesses the ability to substitute known values of sine and calculate the corresponding cosine value.