Question

Find all solutions of the equation in the interval [0, 2π). cos x + sin x tan x = 2

Answer 1

Answer:

The solutions of the equation in the interval [0,2π )

                    ={  $\frac{\pi }{3}$ }

General solution    θ = 2 nπ +α

                                θ =   $2n\pi + \frac{\pi }{3}$

Step-by-step explanation:

Step(i):-

Given equation

                      cos x + sin x tan x = 2

               ⇒      $cos x + sin x \frac{sin x}{cos x} = 2$

    On simplification , we get

              ⇒     $\frac{sin^{2} x+ cos^2x}{cos x} = 2$

we know that trigonometry formula

$sin^{2} x+ cos^2 x = 1$

now we get

             $\frac{1}{cos x} = 2$

⇒         $cos x = \frac{1}{2}$

⇒         cos x = cos 60°

Step(ii):-

General solution of cosθ = cosα

General solution    θ = 2 nπ +α

                               θ = 2 nπ +60°

                                θ =   $2n\pi + \frac{\pi }{3}$

put n = 0 ⇒ θ = 60°

Put n =1 ⇒  θ = 360°+60°= 420°

.....and so on

The solutions of the equation in the interval  =$\frac{\pi }{3}$

Final answer:-

The solutions of the equation in the interval [0,2π )

                    ={  $\frac{\pi }{3}$ }

General solution    θ = 2 nπ +α

                                θ =   $2n\pi + \frac{\pi }{3}$