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

Problem

cos(arctan(12/5))

Solution

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

2. Relate the tangent ratio to the sides of a right triangle, where tan(θ)=opposite/adjacent

3. Assign the side lengths as opposite=12 and adjacent=5

4. Calculate the hypotenuse r using the Pythagorean theorem r=√(,5+12)

5. Simplify the square root to find r=√(,25+144)=√(,169)=13

6. Determine the cosine value using the ratio cos(θ)=adjacent/hypotenuse

7. Substitute the known values to get cos(θ)=5/13

Final Answer

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

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