Find the Exact Value tan(arccos(-4/5))

Problem

tan(arccos(−4/5))

Solution

1. Identify the inner expression as an angle θ=arccos(−4/5)

2. Determine the range of the inverse cosine function, which is [0,π] Since the argument −4/5 is negative, θ must be in the second quadrant (Quadrant II).

3. Use the definition of cosine in a right triangle or on the unit circle, where cos(θ)=x/r Here, x=−4 and r=5

4. Calculate the missing ycoordinate using the Pythagorean identity x2+y2=r2

(−4)2+y2=5

16+y2=25

y2=9

5. Select the positive root y=3 because the sine (and ycoordinate) is positive in Quadrant II.

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

tan(θ)=3/(−4)

Final Answer

tan(arccos(−4/5))=−3/4

Want more problems? Check here!