Question

Solve the trigonometric equation:-

3cos^2x - 2 root 3 sinx cosx - 3sin^2x = 0

Answer 1

$$3 \cos^ {2} (x) - 2 \sqrt {3} \sin (x) \cos (x) - 3 \sin^ {2} (x) = 0$$

Solution:

$$3 \left(\cos^ {2} (x) - \sin^ {2} (x)\right) - 2 \sqrt {3} \sin (x) \cos (x) = 0$$ $$3 \cos (2 x) - \sqrt {3} \sin (2 x) = 0$$ $$3 \cos (2 x) = \sqrt {3} \sin (2 x)$$ $$\cos 2 x / \sin 2 x = \sqrt {3} / 3$$ $$c t g 2 x = \sqrt {3} / 3$$ $$\sqrt {3} / 3 = 1 / \sqrt {3} = \pi / 3$$ $$c t g 2 x = (c t g ^ {2} (x) - 1) / 2 c t g (x), \quad x = \pi n / 2, n \in \mathbb {Z}$$

Answer:

$$x = \pi n / 2 - \pi / 3, n \in \mathbb {Z}$$