Find the Exact Value sin(arccos(3/5))

Problem

sin(arccos(3/5))

Solution

1. Identify the inner expression as an angle θ=arccos(3/5)

2. Apply the definition of the inverse cosine function, which implies cos(θ)=3/5 where 0≤θ≤π

3. Use the Pythagorean identity sin2(θ)+cos2(θ)=1 to find the value of sin(θ)

4. Substitute the known value of cos(θ) into the identity:

sin2(θ)+(3/5)2=1

5. Solve for sin(θ)

sin2(θ)+9/25=1

sin2(θ)=1−9/25

sin2(θ)=16/25

6. Determine the sign of the result. Since the range of arccos(x) is [0,π] and the sine function is non-negative in this interval, we take the positive square root:

sin(θ)=√(,16/25)

sin(θ)=4/5

Final Answer

sin(arccos(3/5))=4/5

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