Find the Exact Value tan(arcsin(-24/26))

Problem

tan(arcsin(−24/26))

Solution

1. Simplify the fraction inside the inverse sine function by dividing the numerator and denominator by their greatest common divisor, 2.

arcsin(−24/26)=arcsin(−12/13)

2. Define an angle θ such that θ=arcsin(−12/13) By the definition of the inverse sine function, sin(θ)=−12/13 where −π/2≤θ≤π/2

3. Identify the quadrant of θ Since the sine value is negative, θ must lie in Quadrant IV, where cos(θ) is positive and tan(θ) is negative.

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

cos2(θ)+(−12/13)2=1

cos2(θ)+144/169=1

cos2(θ)=1−144/169

cos2(θ)=25/169

cos(θ)=5/13

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

tan(θ)=(−12/13)/5/13

tan(θ)=−12/5

Final Answer

tan(arcsin(−24/26))=−12/5

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