Find the Exact Value cos((19pi)/4)

Problem

cos((19*π)/4)

Solution

1. Find the coterminal angle by subtracting multiples of 2*π (which is (8*π)/4 from the given angle to bring it within the range [0,2*π]

(19*π)/4−2*π=(11*π)/4

(11*π)/4−2*π=(3*π)/4

2. Identify the quadrant for the angle (3*π)/4 Since π/2<(3*π)/4<π the angle is in Quadrant II.

3. Determine the reference angle for (3*π)/4 in Quadrant II using the formula π−θ

π−(3*π)/4=π/4

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

cos((3*π)/4)=−cos(π/4)

5. Evaluate the exact value of the cosine of the reference angle.

cos(π/4)=√(,2)/2

6. Combine the results to find the final value.

cos((19*π)/4)=−√(,2)/2

Final Answer

cos((19*π)/4)=−√(,2)/2

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