Find the Exact Value cos(82)

Problem

cos(82^∘)

Solution

1. Identify the given expression as the cosine of an angle measured in degrees, cos(82^∘)

2. Determine if the angle is a standard reference angle. The angle 82^∘ is not one of the common angles (like 30^∘ 45^∘ or 60^∘ found on the unit circle.

3. Check for sum or difference identities. While 82^∘ can be written as 45^∘+37^∘ or 120^∘−38^∘ these do not involve standard angles that yield exact radical values.

4. Conclude that without further context or specific identities (like half-angle formulas for 164^∘, the exact value of cos(82^∘) is typically represented in its trigonometric form or approximated. However, using the cofunction identity cos(θ)=sin(90^∘−θ) we can rewrite it.

5. Apply the cofunction identity.

cos(82^∘)=sin(90^∘−82^∘)

cos(82^∘)=sin(8^∘)

Final Answer

cos(82^∘)=cos(82^∘)

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