Find the Exact Value sin(arctan(-3/4))

Problem

sin(arctan(−3/4))

Solution

1. Define the angle θ such that θ=arctan(−3/4) By the definition of the arctangent function, this means tan(θ)=−3/4 where −π/2<θ<π/2

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

3. Relate the tangent ratio to the sides of a right triangle. In Quadrant IV, we can let the opposite side y=−3 and the adjacent side x=4

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

r=√(,4+(−3)2)

r=√(,16+9)

r=5

5. Determine the value of sin(θ) using the ratio y/r

sin(θ)=(−3)/5

Final Answer

sin(arctan(−3/4))=−3/5

Want more problems? Check here!