Find the Exact Value tan(arcsin(4/7))

Problem

tan(arcsin(4/7))

Solution

1. Identify the inner function as an angle θ=arcsin(4/7) which implies sin(θ)=4/7 where θ is in the interval [−π/2,π/2]

2. Relate the sine value to a right triangle where the opposite side is 4 and the hypotenuse is 7

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

4+b2=7

16+b2=49

b2=33

b=√(,33)

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

tan(θ)=opposite/adjacent

tan(θ)=4/√(,33)

5. Rationalize the denominator by multiplying the numerator and denominator by √(,33)

4/√(,33)⋅√(,33)/√(,33)=(4√(,33))/33

Final Answer

tan(arcsin(4/7))=(4√(,33))/33

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