Find the Exact Value sin(arccos(3/4))

Problem

sin(arccos(3/4))

Solution

1. Identify the inner expression as an angle θ=arccos(3/4) which implies cos(θ)=3/4 where 0≤θ≤π

2. Relate the trigonometric functions using the Pythagorean identity sin2(θ)+cos2(θ)=1

3. Substitute the known value of cos(θ) into the identity to solve for sin(θ)

sin2(θ)+(3/4)2=1

4. Simplify the equation by squaring the fraction and subtracting it from both sides.

sin2(θ)=1−9/16

sin2(θ)=7/16

5. Solve for sin(θ) by taking the square root. Since the range of arccos() is [0,π] the sine of the angle must be non-negative.

sin(θ)=√(,7/16)

sin(θ)=√(,7)/4

Final Answer

sin(arccos(3/4))=√(,7)/4

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