Find the Exact Value sin(arctan(-5/12))

Problem

sin(arctan(−5/12))

Solution

1. Identify the inner function as an angle θ=arctan(−5/12) By the definition of the inverse tangent function, this means tan(θ)=−5/12 where θ must lie in the interval (−π/2,π/2)

2. Determine the quadrant of the angle. Since the tangent value is negative, θ must be in the fourth quadrant (Quadrant IV), where sin(θ) is negative and cos(θ) is positive.

3. Relate the tangent ratio to the sides of a right triangle. Let the opposite side be y=−5 and the adjacent side be x=12

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

r=√(,12+(−5)2)

r=√(,144+25)

r=√(,169)

r=13

5. Evaluate the sine of the angle using the ratio sin(θ)=y/r

sin(θ)=(−5)/13

Final Answer

sin(arctan(−5/12))=−5/13

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