Find the Exact Value cos(pi/10)

Problem

cos(π/10)

Solution

1. Identify the angle in degrees to recognize its relationship to known values.

π/10=18^∘

2. Set up an equation using a multiple of the angle. Let θ=18^∘ Then 5*θ=90^∘

2*θ=90^∘−3*θ

3. Apply the sine function to both sides of the equation.

sin(2*θ)=sin(90^∘−3*θ)

4. Use trigonometric identities for double angles and co-functions.

sin(2*θ)=cos(3*θ)

5. Expand both sides using the double-angle formula for sine and the triple-angle formula for cosine.

2*sin(θ)*cos(θ)=4*cos^3(θ)−3*cos(θ)

6. Divide by cos(θ) since cos(18^∘)≠0

2*sin(θ)=4*cos^2(θ)−3

7. Substitute cos^2(θ)=1−sin^2(θ) to create a quadratic equation in terms of sin(θ)

2*sin(θ)=4*(1−sin^2(θ))−3

8. Rearrange the equation into standard quadratic form.

4*sin^2(θ)+2*sin(θ)−1=0

9. Solve for sin(θ) using the quadratic formula. Since 18^∘ is in the first quadrant, sin(θ)>0

sin(18^∘)=(−2+√(,4−4*(4)*(−1)))/(2*(4))

sin(18^∘)=(√(,5)−1)/4

10. Calculate cos(18^∘) using the Pythagorean 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(π/10)=√(,10+2√(,5))/4

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