Find the Exact Value sin(pi/9)

Problem

sin(π/9)

Solution

1. Identify the expression as the sine of 20 since π/9 radians is equal to 180/9=20

2. Recognize that sin(20) does not have a simple radical form using square roots alone, as it is related to the roots of a cubic equation derived from the triple angle formula.

3. Apply the triple angle identity sin(3*θ)=3*sin(θ)−4*sin3(θ) with θ=π/9

4. Substitute the values into the identity: sin(3⋅π/9)=sin(π/3)=√(,3)/2

5. Formulate the cubic equation 3*x−4*x3=√(,3)/2 where x=sin(π/9)

6. Conclude that while the value can be expressed using roots of this cubic equation or complex numbers via Euler's formula, it is standard in trigonometry to leave sin(π/9) in its transcendental form unless a decimal approximation or complex representation is requested.

Final Answer

sin(π/9)=sin(π/9)

Want more problems? Check here!