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tan x - √3 = 0

We will discuss about the general solution of the equation tan x minus square root of 3 equals 0 (i.e., tan x - √3 = 0) or tan x equals square root of 3 (i.e., tan x = √3).

How to find the general solution of the trigonometric equation tan x = √3 or tan x - √3 = 0?

Solution:

We have,

tan x - √3 = 0

⇒ tan x = √3

⇒ tan x = π3

Again, tan x = √3

⇒ tan x = π3

⇒ tan x = (π + π3)

⇒ tan x = tan 4π3

Let O be the centre of a unit circle. We know that in unit circle, the length of the circumference is 2π.

If we started from A and moves in anticlockwise direction then at the points A, B, A', B' and A, the arc length travelled are 0, π2, π, 3π2, and 2π.

Therefore, from the above unit circle it is clear that the final arm OP of the angle θ lies either in the first or in the final third quadrant.

If the final arm OP lies the first quadrant then,

tan x = √3

⇒ tan x = cos π3

⇒ tan x = ten (2nπ + π3), Where n ∈ I (i.e., n = 0, ± 1, ± 2, ± 3,…….)            

Therefore, x = 2nπ + π3 …………….. (i)

Again, the final arm OP lies in the third quadrant then,

tan x = √3

⇒ tan x = cos 4π3

⇒ tan x = ten (2nπ + 4π3) , Where n ∈ I (i.e., n = 0, ± 1, ± 2, ± 3,…….)

Therefore, x = 2nπ + π3 …………….. (ii)

Therefore, the general solution of equation tan x - √3 = 0 are the infinite sets of values of x given in (i) and (ii).

Hence general solution of tan x - √3 = 0 is x = nπ + π3, n ∈ I.

 Trigonometric Equations




11 and 12 Grade Math

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