Question

Solve the equation (3ain(2x)sin(x))=(3cos(x)).

a) x=0,π
b) x=π/2,3π/2
c) x=π/4,3π/4
d) x=π/6,5π/6

Answer 1

To solve the equation (3sin(2x)sin(x))=(3cos(x)), use the trigonometric identity sin(2x) = 2sin(x)cos(x) to simplify the equation. Then, solve for sin(x) and determine the values of x in the given interval [0, 2π] that satisfy the equation.

Step-by-step explanation:

To solve the equation (3sin(2x)sin(x))=(3cos(x)), we can use the trigonometric identity sin(2x) = 2sin(x)cos(x). We substitute this identity into the equation, giving us 3(2sin(x)cos(x))sin(x) = 3cos(x). Simplifying, we get 6sin^2(x)cos(x) = 3cos(x). Dividing both sides by cos(x), we get 6sin^2(x) = 3. Dividing by 3, we get 2sin^2(x) = 1. Taking the square root of both sides, we get sin(x) = ±1/√2. In the interval [0, 2π], sin(x) is equal to 1/√2 at x = π/4 and 3π/4, and equal to -1/√2 at x = 5π/4 and 7π/4. Therefore, the solutions to the equation are x = π/4, 3π/4, 5π/4, and 7π/4.