Find the Exact Value cot(arccos(-15/17))

Problem

cot(arccos(−15/17))

Solution

1. Identify the inner expression as an angle θ=arccos(−15/17) By the definition of the inverse cosine function, cos(θ)=−15/17 where 0≤θ≤π

2. Determine the quadrant of θ Since the cosine value is negative, θ must be in Quadrant II.

3. Use the Pythagorean identity sin2(θ)+cos2(θ)=1 to find sin(θ)

sin2(θ)+(−15/17)2=1

sin2(θ)+225/289=1

sin2(θ)=1−225/289

sin2(θ)=64/289

4. Solve for sin(θ) Since θ is in Quadrant II, the sine value must be positive.

sin(θ)=√(,64/289)

sin(θ)=8/17

5. Apply the definition of the cotangent function, cot(θ)=cos(θ)/sin(θ)

cot(θ)=(−15/17)/8/17

6. Simplify the fraction by canceling the common denominator.

cot(θ)=−15/8

Final Answer

cot(arccos(−15/17))=−15/8

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