Find the Exact Value tan(arcsin(5/8))

Problem

tan(arcsin(5/8))

Solution

1. Identify the inner function as an angle θ=arcsin(5/8) which implies sin(θ)=5/8 where θ is in the interval [−π/2,π/2]

2. Relate the sine value to a right triangle where the side opposite to θ is 5 and the hypotenuse is 8

3. Calculate the adjacent side b using the Pythagorean theorem a2+b2=c2

5+b2=8

25+b2=64

b2=39

b=√(,39)

4. Determine the value of tan(θ) using the ratio of the opposite side to the adjacent side.

tan(θ)=opposite/adjacent

tan(θ)=5/√(,39)

5. Rationalize the denominator by multiplying the numerator and denominator by √(,39)

5/√(,39)⋅√(,39)/√(,39)=(5√(,39))/39

Final Answer

tan(arcsin(5/8))=(5√(,39))/39

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