Find the Exact Value cos(67.5)

Problem

cos(67.5)

Solution

1. Identify the angle as a half-angle of a common reference angle. Since 67.5×2=135 we can use the half-angle identity for cosine.

2. Apply the formula for the half-angle identity: cos(θ/2)=±√(,(1+cos(θ))/2)

3. Determine the sign based on the quadrant. Since 67.5 is in the first quadrant, the cosine value must be positive.

4. Substitute θ=135 into the formula.

cos(67.5)=√(,(1+cos(135))/2)

5. Evaluate the cosine of 135 Since 135 is in the second quadrant with a reference angle of 45 cos(135)=−√(,2)/2

cos(67.5)=√(,(1−√(,2)/2)/2)

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

cos(67.5)=√(,(2−√(,2))/4)

7. Simplify the radical by taking the square root of the denominator.

cos(67.5)=√(,2−√(,2))/2

Final Answer

cos(67.5)=√(,2−√(,2))/2

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