Find the Exact Value tan(arcsin(-3/5))

Problem

tan(arcsin(−3/5))

Solution

1. Identify the inner function θ=arcsin(−3/5) By definition, this means sin(θ)=−3/5 where −π/2≤θ≤π/2

2. Determine the quadrant of θ Since the sine value is negative and the range of arcsin() is restricted to the first and fourth quadrants, θ must be in the fourth quadrant (Q*I*V.

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

cos2(θ)+(−3/5)2=1

cos2(θ)+9/25=1

cos2(θ)=16/25

4. Solve for cos(θ) Since θ is in Q*I*V the cosine value must be positive.

cos(θ)=√(,16/25)

cos(θ)=4/5

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

tan(θ)=(−3/5)/(4/5)

tan(θ)=−3/4

Final Answer

tan(arcsin(−3/5))=−3/4

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