Find the Exact Value tan(arcsin(8/9))

Problem

tan(arcsin(8/9))

Solution

1. Identify the inner expression as an angle θ=arcsin(8/9) which implies sin(θ)=8/9 for θ in the interval [−π/2,π/2]

2. Relate the sine value to the sides of a right triangle where the opposite side is 8 and the hypotenuse is 9

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

a2+8=9

a2+64=81

a2=17

a=√(,17)

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

tan(θ)=opposite/adjacent

tan(θ)=8/√(,17)

5. Rationalize the denominator by multiplying the numerator and denominator by √(,17)

8/√(,17)⋅√(,17)/√(,17)=(8√(,17))/17

Final Answer

tan(arcsin(8/9))=(8√(,17))/17

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