Find the Exact Value cos(112)

Problem

cos(112^∘)

Solution

1. Identify the given angle as 112^∘ Since this is not a standard reference angle, we check if the task implies radians or degrees. Given the context of "Exact Value" for such a number, it is treated as 112^∘

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

3. Substitute θ=224^∘ into the formula, so θ/2=112^∘

4. Determine the sign. Since 112^∘ is in the second quadrant, the cosine value must be negative.

5. Calculate the exact value using cos(224^∘) Note that 224^∘=180^∘+44^∘ so cos(224^∘)=−cos(44^∘)

6. Simplify the expression to its radical form.

cos(112^∘)=−√(,(1+cos(224^∘))/2)

cos(112^∘)=−√(,(1−cos(44^∘))/2)

Final Answer

cos(112^∘)=−√(,(1−cos(44^∘))/2)

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