Find the Exact Value tan(18)

Identify the relationship between 18 and 90 by letting θ=18 which implies 5*θ=90

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)

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

Expand using double-angle and triple-angle formulas to get 2*sin(θ)*cos(θ)=4*cos3(θ)−3*cos(θ)

Divide by cos(θ) (since cos(18)≠0 to obtain 2*sin(θ)=4*cos2(θ)−3

Substitute cos2(θ)=1−sin2(θ) to form a quadratic equation in terms of sin(θ) 4*sin2(θ)+2*sin(θ)−1=0

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).

Calculate cos(18) using the identity cos(θ)=√(,1−sin2(θ)) which yields cos(18)=√(,1−((√(,5)−1)/4)2)=√(,10+2√(,5))/4

Determine tan(18) by using the ratio sin(18)/cos(18)

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

Find the Exact Value tan(18)

Problem

Solution

Final Answer