Answer:
The solutions of the equation in the interval [0,2π )
={ [tex]\frac{\pi }{3}[/tex] }
General solution θ = 2 nπ +α
θ = [tex]2n\pi + \frac{\pi }{3}[/tex]
Step-by-step explanation:
Step(i):-
Given equation
cos x + sin x tan x = 2
⇒ [tex]cos x + sin x \frac{sin x}{cos x} = 2[/tex]
On simplification , we get
⇒ [tex]\frac{sin^{2} x+ cos^2x}{cos x} = 2[/tex]
we know that trigonometry formula
[tex]sin^{2} x+ cos^2 x = 1[/tex]
now we get
[tex]\frac{1}{cos x} = 2[/tex]
⇒ [tex]cos x = \frac{1}{2}[/tex]
⇒ cos x = cos 60°
Step(ii):-
General solution of cosθ = cosα
General solution θ = 2 nπ +α
θ = 2 nπ +60°
θ = [tex]2n\pi + \frac{\pi }{3}[/tex]
put n = 0 ⇒ θ = 60°
Put n =1 ⇒ θ = 360°+60°= 420°
.....and so on
The solutions of the equation in the interval =[tex]\frac{\pi }{3}[/tex]
Final answer:-
The solutions of the equation in the interval [0,2π )
={ [tex]\frac{\pi }{3}[/tex] }
General solution θ = 2 nπ +α
θ = [tex]2n\pi + \frac{\pi }{3}[/tex]
