Answer:
The exact value of 2 sin(120°) cos(120°) is -√3/2
Explanation:
* Lets revise the trigonometry functions of the double angle
# sin(2x) = 2 sin(x) cos(x)
# cos(2x) = cos²(x) - sin²(x) OR
cos(2x) = 2 cos²(x) - 1 OR
cos(2x) = 1 - 2 sin²(x)
# tan(2x) = 2 tan(x)/(1 - tan²(x))
* Now lets solve the problem
∵ 2 sin(120°) cos(120°)
- Put sin(120°) = sin(2×60°)
∵ sin(2x) = 2 sin(x) cos(x)
∴ sin(120°) = 2 sin(60°) cos(60°)
∵ sin(60°) = √3/2 and cos(60°) = 1/2
∴ sin(120°) = 2 (√3/2) (1/2) = √3/2
∴ sin(120°) = √3/2 ⇒ (1)
- Put cos(120°) = cos(2×60°)
∵ cos(2x) = cos²(x) - sin²(x)
∴ cos(120°) = cos²(60°) - sin²(60°)
∵ cos(60°) = 1/2 and sin(60°) = √3/2
∴ cos(120°) = (1/2)² - (√3/2)² = 1/4 - 3/4 = -2/4 = -1/2
∴ cos(120°) = -1/2 ⇒ (2)
- Substitute (1) and (2) in the expression 2 sin(120) cos(120)
∴ 2 sin(120°) cos(120°) = 2 (√3/2) (-1/2) = -√3/2
* The exact value of 2 sin(120°) cos(120°) is -√3/2