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

Problem

tan(arcsin(3/5))

Solution

1. Identify the inner function as an angle θ where θ=arcsin(3/5)

2. Rewrite the expression using the definition of the inverse sine function, which implies sin(θ)=3/5 for θ in the interval [−π/2,π/2]

3. Relate the sine value to a right triangle where the opposite side is 3 and the hypotenuse is 5

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

3+b2=5

9+b2=25

b2=16

b=4

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

tan(θ)=opposite/adjacent

tan(θ)=3/4

Final Answer

tan(arcsin(3/5))=3/4

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