Simplify cos(22.5)

Problem

cos(22.5)

Solution

1. Identify the angle as a half-angle of a known special angle, since 22.5=45/2

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

3. Determine the sign based on the quadrant. Since 22.5 is in the first quadrant, the cosine value is positive.

4. Substitute θ=45 into the formula.

cos(22.5)=√(,(1+cos(45))/2)

5. Substitute the value of cos(45)=√(,2)/2

cos(22.5)=√(,(1+√(,2)/2)/2)

6. Simplify the expression inside the square root by finding a common denominator in the numerator.

cos(22.5)=√(,(2+√(,2))/2/2)

7. Divide by 2 to reach the final simplified form.

cos(22.5)=√(,(2+√(,2))/4)

cos(22.5)=√(,2+√(,2))/2

Final Answer

cos(22.5)=√(,2+√(,2))/2

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