Question

Solve: $\sin x - \cos x = 0$?

Answer 1

Hint: In the given question, we are required to find all the possible values of $\theta $ that satisfy the given trigonometric equation $\sin x - \cos x = 0$ . For solving such types of questions where we have to solve trigonometric equations, we need to have basic knowledge of algebraic rules and identities as well as a strong grip on trigonometric formulae and identities. We will first transpose one term to the other side of the equation and convert the trigonometric functions into a form of tangent function to solve the trigonometric equation.

Complete step by step answer:
We have to solve the given trigonometric equation $\sin x - \cos x = 0$ .
Shifting the cosine term to right side of the equation, we get,
$ \Rightarrow \sin x = \cos x$
Dividing both sides of the equation by $\cos x$, we get,
$ \Rightarrow \dfrac{{\sin x}}{{\cos x}} = \dfrac{{\cos x}}{{\cos x}}$
Cancelling the common terms in numerator and denominator, we get,
$ \Rightarrow \dfrac{{\sin x}}{{\cos x}} = 1$
Using the trigonometric formula $\tan x = \dfrac{{\sin x}}{{\cos x}}$, we get,
$ \Rightarrow \tan x = 1$
We know that the general solution for the equation $\tan \left( \theta \right) = \tan \left( \phi \right)$ is $\theta = n\pi + \phi $. So, first we have to convert the $\tan x = 1$ into the $\tan \left( \theta \right) = \tan \left( \phi \right)$ form.
Now, we know that the value of the tangent function for the angle $\left( {\dfrac{\pi }{4}} \right)$ is one. So, we get,
$ \Rightarrow \tan x = \tan \left( {\dfrac{\pi }{4}} \right)$
Hence, we have $x = n\pi + \dfrac{\pi }{4}$ .
So the possible values of $x$ for $\sin x - \cos x = 0$ are $n\pi + \dfrac{\pi }{4}$ where n is any integer.

Note:
 Such trigonometric equations can be solved by various methods by applying suitable trigonometric identities and formulae. The general solution of a given trigonometric solution may differ in form, but actually represents the correct solutions. The different forms of general equations are interconvertible into each other.