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

Problem

sin(arctan(−4/3))

Solution

1. Identify the inner expression as an angle θ=arctan(−4/3) This implies that tan(θ)=−4/3

2. Determine the range of the arctangent function. Since the range of arctan(x) is (−π/2,π/2) and the argument −4/3 is negative, θ must lie in the fourth quadrant (Quadrant IV).

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

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

r=√(,3+(−4)2)

r=√(,9+16)

r=5

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

sin(θ)=(−4)/5

Final Answer

sin(arctan(−4/3))=−4/5

Want more problems? Check here!