Find the Exact Value sin(arccos(3/7))

Problem

sin(arccos(3/7))

Solution

1. Identify the inner expression as an angle θ=arccos(3/7) which implies cos(θ)=3/7 where 0≤θ≤π

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

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

sin2(θ)+(3/7)2=1

4. Solve for sin2(θ) by squaring the fraction and subtracting it from 1:

sin2(θ)+9/49=1

sin2(θ)=1−9/49

sin2(θ)=40/49

5. Take the square root of both sides, noting that since θ=arccos(3/7) is in the interval [0,π] sin(θ) must be positive:

sin(θ)=√(,40/49)

6. Simplify the radical expression:

sin(θ)=√(,4⋅10)/7

sin(θ)=(2√(,10))/7

Final Answer

sin(arccos(3/7))=(2√(,10))/7

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