Find the Exact Value cos(10)

Problem

cos(10^∘)

Solution

1. Identify the task as finding the exact value of the cosine of 10 degrees.

2. Recognize that 10^∘ is not a standard angle (like 30^∘ 45^∘ or 60^∘ with a simple radical value.

3. Relate the angle to a known triple-angle identity, specifically cos(3*θ)=4*cos^3(θ)−3*cos(θ)

4. Substitute θ=10^∘ into the identity to get cos(30^∘)=4*cos^3(10^∘)−3*cos(10^∘)

5. Evaluate cos(30^∘)=√(,3)/2 and set up the cubic equation 4*x^3−3*x−√(,3)/2=0 where x=cos(10^∘)

6. Determine that the exact value involves roots of a cubic equation, which can be expressed using complex numbers or nested radicals, but is typically left in its trigonometric form unless a specific algebraic form is required.

Final Answer

cos(10^∘)=cos(10^∘)

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