Find the Exact Value tan(arcsin(-12/13))

Problem

tan(arcsin(−12/13))

Solution

1. Identify the inner function and let θ=arcsin(−12/13) This implies sin(θ)=−12/13

2. Determine the range of the inverse sine function. Since the range of arcsin(x) is [−π/2,π/2] and the sine value is negative, θ must be in the fourth quadrant (Quadrant IV).

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

cos2(θ)+(−12/13)2=1

4. Solve for cos(θ) Since θ is in Quadrant IV, the cosine value must be positive.

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, which is tan(θ)=sin(θ)/cos(θ)

tan(θ)=(−12/13)/5/13

6. Simplify the fraction to find the final value.

tan(θ)=−12/5

Final Answer

tan(arcsin(−12/13))=−12/5

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