Find the Exact Value tan(arcsin(-5/13))

Problem

tan(arcsin(−5/13))

Solution

1. Identify the inner expression as an angle θ=arcsin(−5/13) By definition, this means sin(θ)=−5/13 where −π/2≤θ≤π/2

2. Determine the quadrant of θ Since the sine value is negative, θ must lie in Quadrant IV. In this quadrant, cos(θ) is positive and tan(θ) is negative.

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

cos2(θ)+(−5/13)2=1

cos2(θ)+25/169=1

cos2(θ)=144/169

cos(θ)=12/13

4. Apply the definition of the tangent function, tan(θ)=sin(θ)/cos(θ) to find the final value.

tan(θ)=(−5/13)/(12/13)

tan(θ)=−5/12

Final Answer

tan(arcsin(−5/13))=−5/12

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