Question

Find all solutions of the equation in the interval [0, 2). (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.) 6 cos x + 6 sin x tan x = 12

Answer 1

To solve the equation 6 cos x + 6 sin x tan x = 12 on the interval [0, 2π), we utilize trigonometric identities to simplify the equation to cos x = 1/2, which has solutions at x = π/3 and x = 5π/3.

Step-by-step explanation:

The question presented asks to find all solutions of the equation 6 cos x + 6 sin x tan x = 12 in the interval [0, 2π). To solve this equation, we can first simplify it by factoring out the common factor of 6:

6(cos x + sin x tan x) = 12

Dividing both sides by 6, we get:

cos x + sin x tan x = 2

Using the trigonometric identity that tan x = sin x / cos x, we can rewrite our equation as:

cos x + sin2 x / cos x = 2

Multiplying through by cos x gives:

cos2 x + sin2 x = 2 cos x

Now, applying the Pythagorean identity sin2 x + cos2 x = 1, our equation simplifies to:

1 = 2 cos x

Solving for cos x, we get:

cos x = 1/2

The solutions for cos x = 1/2 in the interval [0, 2π) are x = π/3 and x = 5π/3. Thus, these are the solutions we're looking for.