Simplify 2cos(165)^2-1

Problem

2*cos^2(165^∘)−1

Solution

1. Identify the expression as a form of the double-angle identity for cosine.

2. Recall the identity cos(2*θ)=2*cos^2(θ)−1

3. Substitute θ=165^∘ into the identity to rewrite the expression.

2*cos^2(165^∘)−1=cos(2⋅165^∘)

4. Multiply the angle inside the cosine function.

cos(2⋅165^∘)=cos(330^∘)

5. Determine the reference angle for 330^∘ which is in the fourth quadrant.

360^∘−330^∘=30^∘

6. Evaluate the cosine of 330^∘ using the unit circle, noting that cosine is positive in the fourth quadrant.

cos(330^∘)=cos(30^∘)=√(,3)/2

Final Answer

2*cos^2(165^∘)−1=√(,3)/2

Want more problems? Check here!