Find the Exact Value tan(18)

Problem

tan(18^∘)

Solution

1. Identify the relationship between 18^∘ and 90^∘ by letting θ=18^∘ which implies 5*θ=90^∘

2. Split the equation into 2*θ=90^∘−3*θ and apply the sine function to both sides to get sin(2*θ)=sin(90^∘−3*θ)

3. Apply trigonometric identities to rewrite the equation as sin(2*θ)=cos(3*θ)

4. Expand using double-angle and triple-angle formulas to get 2*sin(θ)*cos(θ)=4*cos^3(θ)−3*cos(θ)

5. Divide by cos(θ) (since cos(18^∘)≠0 to obtain 2*sin(θ)=4*cos^2(θ)−3

6. Substitute cos^2(θ)=1−sin^2(θ) to form a quadratic equation in terms of sin(θ) 4*sin^2(θ)+2*sin(θ)−1=0

7. Solve the quadratic equation using the quadratic formula to find sin(18^∘)=(−2±√(,4−4*(4)*(−1)))/(2*(4))=(−1+√(,5))/4 (taking the positive root since 18^∘ is in the first quadrant).

8. Calculate cos(18^∘) using the identity cos(θ)=√(,1−sin^2(θ)) which yields cos(18^∘)=√(,1−((√(,5)−1)/4)^2)=√(,10+2√(,5))/4

9. Determine tan(18^∘) by using the ratio sin(18^∘)/cos(18^∘)

10. Simplify the expression (√(,5)−1)/√(,10+2√(,5))

Final Answer

tan(18^∘)=(√(,5)−1)/√(,10+2√(,5))

Want more problems? Check here!