Find the Exact Value cos(74)

Problem

cos(74^∘)

Solution

1. Identify the angle as 74^∘ Since this is not a standard unit circle angle, use a sum or difference formula.

2. Rewrite the angle as a sum of two angles with known trigonometric values, such as 30^∘ and 44^∘ or 45^∘ and 29^∘ However, a more effective approach for an exact value involves the sum formula cos(A+B)=cos(A)*cos(B)−sin(A)*sin(B) using 37^∘+37^∘ or half-angle identities.

3. Apply the half-angle identity cos(θ)=√(,(1+cos(2*θ))/2) or recognize that 74^∘ is 90^∘−16^∘

4. Use the specific exact radical form for cos(74^∘) which is derived from the geometry of a 16−74−90 triangle or complex radical expressions.

5. Simplify the expression to its exact radical form.

Final Answer

cos(74^∘)=(√(,50−10√(,5))−√(,6)−√(,2))/8

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