Question

Solve on the interval [0,2pi] 3(sec x)^2 -4=0

Answer 1

Answer:

pi/6, 5pi/6, 7pi/6, 11pi/6

Step-by-step explanation:

3(sec x)^2 -4=0

Add 4 on both sides:

3(sec x)^2=4

Divide 3 on both sides:

(sec x)^2=4/3

Take square root of both sides:

sec x=plus/minus sqrt(4/3)

Reciprocal identity:

cos x=plus/minus sqrt(3/4)

Simplify the radical:

cos x=plus/minus sqrt(3)/2

So we are looking on the unit circle where the first coordinate of the order pair is either sqrt(3)/2 or -sqrt(3)/2.

This happens at pi/6, 5pi/6, 7pi/6, 11pi/6

Answer 2

Answer:

$\sf \boxed{\sf x = \dfrac{\pi}{6},\dfrac{5\pi}{6},\dfrac{7\pi}{6},\dfrac{11\pi}{6}}$

Step-by-step explanation:

A trigonometric equation is given to us , and we need to find the solutions of the equation within the interval [ 0,2π ]

The given equation is ,

$\sf\longrightarrow 3(sec x)^2-4=0$

This can be written as ,

$\sf\longrightarrow 3sec^2x - 4 = 0$

Add 4 to both sides of the equation ,

$\sf\longrightarrow 3sec^2x = 4$

Divide both sides by 3 ,

$\sf\longrightarrow sec^2x =\dfrac{4}{3}$

Put squareroot on both sides ,

$\sf\longrightarrow sec \ x =\sqrt{\dfrac{4}{3} }$

Simplify ,

$\sf\longrightarrow sec\ x =\pm \dfrac{2}{\sqrt3}$

Multiply numerator and denominator by √3 ,

$\sf\longrightarrow \bf sec(x) = \dfrac{ 2\sqrt3}{3},-\dfrac{2\sqrt3}{3}$

Solve for x ,

$\sf\longrightarrow x = \dfrac{\pi}{6} +2\pi n , \dfrac{11\pi}{6}+2\pi n , \textsf{ for any integer n } .$

Therefore all the possible solutions are ,

$\sf\longrightarrow \boxed{\blue{\sf x = \dfrac{\pi}{6},\dfrac{5\pi}{6},\dfrac{7\pi}{6},\dfrac{11\pi}{6}}}$