Question

How do you find all solutions of the equations $\cos x+\sin x\tan x=2$ in the interval $[0,2\pi )$ ?
(a) By checking random values
(b) Trying drawing the graph
(c) Simplifying the equation
(d) None of these

Answer 1

Hint: We are given the trigonometric equation $\cos x+\sin x\tan x=2$. To get the solution as x, we need to simplify the equation in terms of sinx or cosx. Thus we are getting the value of sinx or cos x as a given number. Thus, we can find the value of x in the general form and that is what we are looking for.

Complete step-by-step answer:
According to the question, we have our equation given as, $\cos x+\sin x\tan x=2$
Now, putting, $\tan x=\dfrac{\sin x}{\cos x}$ , we are getting,
$\Rightarrow \cos x+\sin x.\dfrac{\sin x}{\cos x}=2$
Multiplying both sinx terms in the numerator,
$\Rightarrow \cos x+\dfrac{{{\sin }^{2}}x}{\cos x}=2$
Now again, we know as per the trigonometric identities,${{\sin }^{2}}x=1-{{\cos }^{2}}x$ , because, ${{\sin }^{2}}x+{{\cos }^{2}}x=1$.
So, now if we put the value of ${{\sin }^{2}}x$in our equation, we will get a equation with numbers and cosx.
$\Rightarrow \cos x+\dfrac{1-{{\cos }^{2}}x}{\cos x}=2$
Multiplying both sides with $\cos x$ , we will get rid of the denominator,
$\Rightarrow {{\cos }^{2}}x+1-{{\cos }^{2}}x=2\cos x$
After more simplification and then cancelling out,
$\Rightarrow 2\cos x=1$
So, now, we get the value of $\cos x$ be, $\cos x=\dfrac{1}{2}$ .
Hence, according to the trigonometric table, the value of cosine function in the point $\dfrac{\pi }{3}$ is $\dfrac{1}{2}$ . If we try to use the general solution of cosine function, we will get the solution of x as,
$x=2n\pi \pm \dfrac{\pi }{3},n\in \mathbb{Z}$
As, the general solution of $\cos x=\cos y$, gives us, $x=2n\pi \pm y,n\in \mathbb{Z}$.
So, the correct answer is “Option (c)”.

Note: All possible values of unknown which satisfy the given equation are called solutions of the given equation. For a complete solution “all possible values” satisfying the equation must be obtained. When we try to solve a trigonometric equation, we try to find out all sets of values of the angle, which satisfy the given equation. Sometimes, in simple equations and when it is easy to draw a graph of an equation.