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Question: Solve: \(\sin x - \cos x = 0\)?...

Solve: sin⁡x−cos⁡x=0\sin x - \cos x = 0?

Explanation

Solution

In the given question, we are required to find all the possible values of θ\theta that satisfy the given trigonometric equation sin⁡x−cos⁡x=0\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\sin x - \cos x = 0 .
Shifting the cosine term to right side of the equation, we get,
⇒sin⁡x=cos⁡x\Rightarrow \sin x = \cos x
Dividing both sides of the equation by cos⁡x\cos x, we get,
⇒sin⁡xcos⁡x=cos⁡xcos⁡x\Rightarrow \dfrac{{\sin x}}{{\cos x}} = \dfrac{{\cos x}}{{\cos x}}
Cancelling the common terms in numerator and denominator, we get,
⇒sin⁡xcos⁡x=1\Rightarrow \dfrac{{\sin x}}{{\cos x}} = 1
Using the trigonometric formula tan⁡x=sin⁡xcos⁡x\tan x = \dfrac{{\sin x}}{{\cos x}}, we get,
⇒tan⁡x=1\Rightarrow \tan x = 1
We know that the general solution for the equation tan⁡(θ)=tan⁡(ϕ)\tan \left( \theta \right) = \tan \left( \phi \right) is θ=nπ+ϕ\theta = n\pi + \phi . So, first we have to convert the tan⁡x=1\tan x = 1 into the tan⁡(θ)=tan⁡(ϕ)\tan \left( \theta \right) = \tan \left( \phi \right) form.
Now, we know that the value of the tangent function for the angle (π4)\left( {\dfrac{\pi }{4}} \right) is one. So, we get,
⇒tan⁡x=tan⁡(π4)\Rightarrow \tan x = \tan \left( {\dfrac{\pi }{4}} \right)
Hence, we have x=nπ+π4x = n\pi + \dfrac{\pi }{4} .
So the possible values of xx for sin⁡x−cos⁡x=0\sin x - \cos x = 0 are nπ+π4n\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.