Find the Exact Value tan(arccos(-12/13))

Problem

tan(arccos(−12/13))

Solution

1. Identify the angle θ defined by the inverse cosine function such that θ=arccos(−12/13)

2. Determine the range of the inverse cosine function, which is [0,π] Since the argument −12/13 is negative, θ must be in the second quadrant (Quadrant II).

3. Use the definition of cosine in terms of x y and r where cos(θ)=x/r Here, x=−12 and r=13

4. Calculate the value of y using the Pythagorean identity x2+y2=r2

(−12)2+y2=13

144+y2=169

y2=25

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

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

tan(θ)=5/(−12)

Final Answer

tan(arccos(−12/13))=−5/12

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