Find the Exact Value cos(18)

Problem

cos(18^∘)

Solution

1. Set up the relationship between angles. Let θ=18^∘ Then 5*θ=90^∘ which can be written as 2*θ=90^∘−3*θ

2. Apply the sine function to both sides.

sin(2*θ)=sin(90^∘−3*θ)

3. Use the co-function identity sin(90^∘−x)=cos(x)

sin(2*θ)=cos(3*θ)

4. Expand using double-angle and triple-angle identities.

2*sin(θ)*cos(θ)=4*cos^3(θ)−3*cos(θ)

5. Divide by cos(θ) since cos(18^∘)≠0

2*sin(θ)=4*cos^2(θ)−3

6. Substitute the Pythagorean identity cos^2(θ)=1−sin^2(θ) to get an equation in terms of sin(θ)

2*sin(θ)=4*(1−sin^2(θ))−3

7. Rearrange into a quadratic equation.

4*sin^2(θ)+2*sin(θ)−1=0

8. Solve for sin(θ) using the quadratic formula x=(−b±√(,b^2−4*a*c))/(2*a)

sin(θ)=(−2±√(,4−4*(4)*(−1)))/8

sin(θ)=(−2±√(,20))/8

sin(θ)=(−1±√(,5))/4

9. Select the positive root because 18^∘ is in the first quadrant.

sin(18^∘)=(√(,5)−1)/4

10. Calculate cos(18^∘) using the identity cos(θ)=√(,1−sin^2(θ))

cos(18^∘)=√(,1−((√(,5)−1)/4)^2)

11. Simplify the expression inside the square root.

cos(18^∘)=√(,1−(5−2√(,5)+1)/16)

cos(18^∘)=√(,(16−(6−2√(,5)))/16)

cos(18^∘)=√(,10+2√(,5))/4

Final Answer

cos(18^∘)=√(,10+2√(,5))/4

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