Find the Exact Value cos((19pi)/6)

Problem

cos((19*π)/6)

Solution

1. Find the coterminal angle within the interval [0,2*π) by subtracting multiples of 2*π

(19*π)/6−2*π=(19*π)/6−(12*π)/6

(19*π)/6−(12*π)/6=(7*π)/6

2. Identify the quadrant for the angle (7*π)/6 Since π<(7*π)/6<(3*π)/2 the angle lies in Quadrant III.

3. Determine the reference angle by subtracting π from the angle.

θ′=(7*π)/6−π

θ′=π/6

4. Apply the cosine sign for Quadrant III. In Quadrant III, the cosine function is negative.

cos((7*π)/6)=−cos(π/6)

5. Evaluate the exact value using the known value for cos(π/6)=√(,3)/2

cos((7*π)/6)=−√(,3)/2

Final Answer

cos((19*π)/6)=−√(,3)/2

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