Find the Exact Value cos(-(13pi)/4)

Problem

cos(−(13*π)/4)

Solution

1. Use the even-odd property of the cosine function, which states that cos(−θ)=cos(θ)

cos(−(13*π)/4)=cos((13*π)/4)

2. Find a coterminal angle within the interval [0,2*π) by subtracting multiples of 2*π (which is (8*π)/4.

(13*π)/4−2*π=(13*π)/4−(8*π)/4

(13*π)/4−(8*π)/4=(5*π)/4

3. Determine the reference angle for (5*π)/4 Since this angle is in the third quadrant, the reference angle is θ′=θ−π

θ′=(5*π)/4−(4*π)/4

θ′=π/4

4. Apply the quadrant sign to the cosine value. In the third quadrant, the cosine function is negative.

cos((5*π)/4)=−cos(π/4)

5. Substitute the known value for cos(π/4) which is √(,2)/2

−cos(π/4)=−√(,2)/2

Final Answer

cos(−(13*π)/4)=−√(,2)/2

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