Find the Exact Value sin(arctan(-4))

Problem

sin(arctan(−4))

Solution

1. Identify the inner function as an angle θ=arctan(−4) By the definition of the inverse tangent function, this means tan(θ)=−4 where θ must be in the interval (−π/2,π/2)

2. Determine the quadrant of θ Since the tangent value is negative and the range of arctan() is (−π/2,π/2) θ must lie in Quadrant IV.

3. Represent the tangent ratio as tan(θ)=y/x=(−4)/1 In Quadrant IV, we can let y=−4 and x=1

4. Calculate the hypotenuse r using the Pythagorean theorem r=√(,x2+y2)

r=√(,1+(−4)2)

r=√(,1+16)

r=√(,17)

5. Apply the definition of the sine function, sin(θ)=y/r to find the exact value.

sin(θ)=(−4)/√(,17)

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

sin(θ)=−(4√(,17))/17

Final Answer

sin(arctan(−4))=−(4√(,17))/17

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