Find the Exact Value cos(112.5)

Problem

cos(112.5^∘)

Solution

1. Identify the angle as a half-angle of a known special angle. Since 112.5^∘=225^∘/2 we can use the half-angle identity for cosine.

2. Recall the half-angle formula for cosine:

cos(θ/2)=±√(,(1+cos(θ))/2)

3. Determine the sign of the result. Since 112.5^∘ is in the second quadrant (between 90^∘ and 180^∘, the cosine value must be negative.

cos(112.5^∘)=−√(,(1+cos(225^∘))/2)

4. Find the value of cos(225^∘) The reference angle for 225^∘ is 45^∘ in the third quadrant, where cosine is negative.

cos(225^∘)=−√(,2)/2

5. Substitute the value into the formula:

cos(112.5^∘)=−√(,(1−√(,2)/2)/2)

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

cos(112.5^∘)=−√(,(2−√(,2))/2/2)

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

cos(112.5^∘)=−√(,(2−√(,2))/4)

8. Extract the square root of the denominator.

cos(112.5^∘)=−√(,2−√(,2))/2

Final Answer

cos(112.5^∘)=−√(,2−√(,2))/2

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