Tangent function is defined as the ratio of the side perpendiculardivided by the adjacent.

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Let O be the centre of a unit circle. We know that in unit circle, the length of the circumference is 2π. tan θ = 0

If we started from A and moves in anticlockwise direction then at the points A, B, A", B" và A, the arc length travelled are 0, (fracπ2), π, (frac3π2), & 2π.

tan θ = (fracPMOM)

Now, tan θ = 0

⇒ (fracPMOM) = 0

⇒ PM = 0.

So when will the tangent be equal khổng lồ zero?

Clearly, if PM = 0 then the final arm OP of the angle θcoincides with OX or OX".

Similarly, the final arm OPcoincides with OX or OX" when θ = π, 2π, 3π, 4π, ……….. , - π, -2π, -3π,-4π, ……….. I.e. When θ an integral multiples of π i.e., when θ = nπ where n ∈Z (i.e., n = 0, ± 1, ± 2, ± 3,…….)

Hence, θ = nπ, n ∈Z is the general solution of the given equation rã θ = 0

1. Find the general solution of the equation chảy 2x = 0

Solution:

tan 2x = 0

⇒ 2x = nπ, where, n = 0, ± 1, ± 2, ± 3, …….

⇒ x = (fracnπ2), where, n = 0, ± 1, ± 2, ± 3, …….

Therefore, the general solution of the trigonometric equation tung 2x = 0 is x = (fracnπ2), where, n = 0, ± 1, ± 2, ± 3, …….

2. Find the general solution of the equation tan (fracx2) = 0

Solution:

tan (fracx2) = 0

⇒ (fracx2) = nπ, where, n = 0, ± 1, ± 2, ± 3, …….

⇒ x = 2nπ, where, n = 0, ± 1, ± 2, ± 3, …….

Therefore, the general solution of the trigonometric equation chảy (fracx2) = 0 is x = 2nπ, where, n = 0, ± 1, ± 2, ± 3, …….

3. What is the general solution of the equation tung x + tan 2x + chảy 3x = tan x tan 2x chảy 3x?

Solution:

tan x + rã 2x + rã 3x = chảy x tung 2x rã 3x

⇒ tan x + chảy 2x = - chảy 3x + rã x rã 2x tan 3x

⇒ chảy x + tan 2x = - tung 3x(1 - tung x tung 2x)

⇒ (fractan x + tan 2x1 - chảy x rã 2x) = - rã 3x

⇒ chảy (x + 2x) = - tung 3x

⇒ tung 3x = - rã 3x

⇒ 2 tan 3x = 0

⇒ rã 3x = 0

⇒ 3x = nπ, where n = 0, ± 1, ± 2, ± 3,…….

x = (fracnπ3), where n = 0, ± 1, ± 2, ± 3,…….

Therefore, the general solution of the trigonometric equation rã x + chảy 2x + chảy 3x = tan x tung 2x tung 3x is x = (fracnπ3), where n = 0, ± 1, ± 2, ± 3,…….

4. Find the general solution of the equation chảy (frac3x4) = 0

Solution:

tan (frac3x4) = 0

⇒ (frac3x4) = nπ, where, n = 0, ± 1, ± 2, ± 3, …….

⇒ x = (frac4nπ3), where, n = 0, ± 1, ± 2, ± 3, …….

Therefore, the general solution of the trigonometric equation tan (frac3x4) = 0 is x = (frac4nπ3), where, n = 0, ± 1, ± 2, ± 3, …….

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Trigonometric Equations