Find the Exact Value sin(arctan(7/24))

Problem

sin(arctan(7/24))

Solution

1. Identify the inner expression as an angle θ=arctan(7/24) which implies tan(θ)=7/24

2. Relate the tangent ratio to the sides of a right triangle, where the opposite side O=7 and the adjacent side A=24

3. Calculate the hypotenuse H using the Pythagorean theorem H=√(,O2+A2)

H=√(,7+24)

H=√(,49+576)

H=√(,625)

H=25

4. Determine the value of sin(θ) using the ratio of the opposite side to the hypotenuse O/H

sin(θ)=7/25

Final Answer

sin(arctan(7/24))=7/25

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