Find the Exact Value tan(arcsin(-3/4))

Problem

tan(arcsin(−3/4))

Solution

1. Identify the inner function as an angle θ=arcsin(−3/4)

2. Determine the range of the inverse sine function, which is [−π/2,π/2] Since the sine value is negative, θ must be in the interval (−π/2,0) which corresponds to the fourth quadrant.

3. Use the definition of sine to relate the sides of a right triangle, where sin(θ)=opposite/hypotenuse=−3/4

4. Assign values to the sides: let the opposite side y=−3 and the hypotenuse r=4

5. Calculate the adjacent side x using the Pythagorean theorem x2+y2=r2

x2+(−3)2=4

x2+9=16

x2=7

x=√(,7)

6. Note that x is positive because the angle is in the fourth quadrant.

7. Apply the definition of tangent, which is tan(θ)=opposite/adjacent=y/x

tan(θ)=(−3)/√(,7)

8. Rationalize the denominator by multiplying the numerator and denominator by √(,7)

(−3)/√(,7)⋅√(,7)/√(,7)=−(3√(,7))/7

Final Answer

tan(arcsin(−3/4))=−(3√(,7))/7

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