Find the Exact Value sin(arctan(5/12))

Problem

sin(arctan(5/12))

Solution

1. Identify the inner expression as an angle θ=arctan(5/12) which implies tan(θ)=5/12

2. Relate the tangent ratio to the sides of a right triangle, where the opposite side O=5 and the adjacent side A=12

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

H=√(,5+12)

H=√(,25+144)

H=√(,169)

H=13

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

sin(θ)=5/13

Final Answer

sin(arctan(5/12))=5/13

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