Find the Exact Value cos((13pi)/8)

Problem

cos((13*π)/8)

Solution

1. Identify the quadrant of the angle. Since (13*π)/8 is between (12*π)/8 (1.5*π and (16*π)/8 (2*π, the angle lies in Quadrant IV, where the cosine function is positive.

2. Find the reference angle. Subtract the angle from 2*π to find the distance to the x-axis.

(θ_ref)=2*π−(13*π)/8

(θ_ref)=(16*π)/8−(13*π)/8=(3*π)/8

3. Apply the half-angle formula for cosine. Use the identity cos(θ/2)=±√(,(1+cos(θ))/2) Let θ/2=(3*π)/8 which means θ=(3*π)/4

cos((3*π)/8)=√(,(1+cos((3*π)/4))/2)

4. Substitute the known value cos((3*π)/4)=−√(,2)/2 into the formula.

cos((3*π)/8)=√(,(1−√(,2)/2)/2)

5. Simplify the expression by multiplying the numerator and denominator inside the square root by 2

cos((3*π)/8)=√(,(2−√(,2))/4)

cos((3*π)/8)=√(,2−√(,2))/2

6. Conclude that since cos((13*π)/8)=cos((3*π)/8) due to the properties of Quadrant IV, the value remains positive.

Final Answer

cos((13*π)/8)=√(,2−√(,2))/2

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