Question

Find principal solutions as well as general solutions of the below given trigonometric equation :—
$\underline{ \boxed{ \quad\sec(x) = 2 \quad }}$

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Answer 1

Given : sec (x) = 2

$\frac{1}{cos (x) } = 2\\\\\frac{1}{2}= cos (x)\\ \\cos(x)= \frac{1}{2}$

We know that cos 60° = $\frac{1}{2}$

We find the value of x where  cos is positive.

cos is positive in $I^s^t$ and $IV ^t^h$ Quadrant

Value in Ist Quadrant = 60°

Value in IVth Quadrant = 360°-60°= 300°

So Principle solution are

x= 60°  and x = 300°

x = 60×$\frac{\pi }{180}$ and x = 300×$\frac{\pi }{180}$

x = $\frac{\pi }{3 }$ and x $= \frac{5\pi }{3}$

To find general solution

Let cos x = cos y           .......(1)

and given cos x = $\frac{1}{2}$       .......(2)

From (1) and (2)

cos y = $\frac{1}{2}$

(calculated cos $\frac{\pi }{3 }$ = $\frac{1}{2}$  while finding prinicipal solutions)

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

⇒ y = $\frac{\pi }{3}$

Since cos x = cos y

General solution is

x = 2n$\pi$ ± y where n ∈ Z

Put y = \frac{\pi }{3}

Hence , x = 2n$\pi$±\frac{\pi }{3} where n ∈ Z

Answer 2

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

PRINCIPAL SOLUTION

It is given that

$\sec(x) = 2$

$\displaystyle \: \implies \: cos \: x \: = \frac{1}{2}$

So here value of cosx is Positive

Therefore x is First Quadrant or Fourth Quadrant

Now when x is in First Quadrant

$\displaystyle \: cosx = cos \frac{\pi}{3}$

$\implies \: \displaystyle \: x = \frac{\pi}{3}$

Again when x is in Fourth Quadrant

$\displaystyle \: cosx = cos \frac{\pi}{3}$

$\implies \: \displaystyle \: cosx = cos \: (2\pi - \frac{\pi}{3} )$

$\implies \: \displaystyle \: cosx = cos \: \frac{5\pi}{3}$

$\implies \: \displaystyle \: x = \: \frac{5\pi}{3}$

Here principal Solution is

$\displaystyle \: x = \frac{\pi}{3} \: \: and \: \: \: \frac{5\pi}{3}$

GENERAL SOLUTION

$\displaystyle \: cosx = \frac{1}{2}$

$\implies \: \displaystyle \: cosx = cos \: \frac{\pi}{3}$

$\implies \: \displaystyle \: x = \: ( \: 2n\pi + \frac{\pi}{3} ) \: \: where \: \: n \in \mathbb{Z}$

Which is the required General Solution