Question

How do you find all solutions for $\sec 3x = - 1$if$0 \leqslant \theta < 2\pi $ ?

Answer 1

Hint: we will solve the above question by finding the value of x which can be done by taking inverse of secant. And after that we will find similarly all the other values of $x$ that are under $0$ to .

Complete step by step answer:
We will start by writing the given expression:
$ \Rightarrow \sec 3x = - 1$
The above expression can also be re write in the form of cosine as:
$ \Rightarrow \dfrac{1}{{\cos 3x}} = - 1$
Now, relocate cox3x to the right side of the equation we get:
$ \Rightarrow 1 = - 1 \times \cos 3x$
Simplify and rewrite the above expression:
$ \Rightarrow \cos 3x = - 1$………………. Eq. $(1)$
The above expression can also be rewrite as:
$ \Rightarrow \cos {180^ \circ } = - 1$
Therefore
$ \Rightarrow 3x = {180^ \circ }$
Divide by $3$ to both the sides of the equation:
$ \Rightarrow \dfrac{{3x}}{3} = \dfrac{{{{180}^ \circ }}}{3}$
Cancel the common factor of $3$:
$ \Rightarrow \dfrac{{{3}x}}{{{3}}} = \dfrac{{{3} \times {{60}^ \circ }}}{{{3}}}$
Simplify and rewrite the equation:
$ \Rightarrow x = {60^ \circ }$
So, for $0 \leqslant \theta < 2\pi $, where $\theta $ is $x$:
Also we can write the above expression as: $0 \leqslant x < 360$
Now for $3x$ the above expression will be:
$ \Rightarrow 0 \leqslant 3x < 3 \times 360$
Simplify and rewrite:
$ \Rightarrow 0 \leqslant 3x < 1080$
Therefore;
$ \Rightarrow \cos (180 + 360) = - 1$
$ \Rightarrow 3x = 540$
Divide by $3$ to both the sides of the equation:
$ \Rightarrow \dfrac{{3x}}{3} = \dfrac{{{{540}^ \circ }}}{3}$
Cancel the common factor of $3$:
$ \Rightarrow \dfrac{{{3}x}}{{{3}}} = \dfrac{{{3} \times {{180}^ \circ }}}{{{3}}}$
Simplify and rewrite the equation:
$ \Rightarrow x = {180^ \circ }$
Similarly;
$ \Rightarrow \cos (540 + 360) = - 1$
$ \Rightarrow 3x = 900$
Divide by $3$ to both the sides of the equation:
$ \Rightarrow \dfrac{{3x}}{3} = \dfrac{{{{900}^ \circ }}}{3}$
Cancel the common factor of $3$:
$ \Rightarrow \dfrac{{{3}x}}{{{3}}} = \dfrac{{{3} \times {{300}^ \circ }}}{{{3}}}$
Simplify and rewrite the equation:
$ \Rightarrow x = {300^ \circ }$

So, the solutions are $x = {60^ \circ },{180^ \circ },{300^ \circ }$.

Note: Range for trigonometry function are given below:
$\sin x = [ - 1,1]$
$\cos x = [ - 1,1]$
$\tan x = R$
$\cot x = R$
$\cos ecx = R - \{ x: - 1 < x < 1\} $
$\sec x = R - \{ x: - 1 < x < 1\} $
Remember we would not stop our solution after finding only one value of $x$, we will continue to calculate the values of $x$ until the value becomes greater than the limit given to us.
And, also to solve these types of questions you should be familiar with all the basic trigonometric formulas.