Find the Exact Value tan(arcsin(6/7))

Problem

tan(arcsin(6/7))

Solution

1. Identify the inner function as an angle θ=arcsin(6/7) which implies sin(θ)=6/7

2. Determine the constraints of the angle θ Since the range of arcsin() is [−π/2,π/2] and the value 6/7 is positive, θ must be in the first quadrant.

3. Relate the sine value to a right triangle where the opposite side is 6 and the hypotenuse is 7

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

a2+6=7

a2+36=49

a2=13

a=√(,13)

5. Evaluate the tangent of the angle θ using the ratio of the opposite side to the adjacent side.

tan(θ)=opposite/adjacent

tan(θ)=6/√(,13)

6. Rationalize the denominator by multiplying the numerator and denominator by √(,13)

6/√(,13)⋅√(,13)/√(,13)=(6√(,13))/13

Final Answer

tan(arcsin(6/7))=(6√(,13))/13

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