tan θ = tan ∝

How to find the general solution of an equation of the form tan θ = tan ∝?

Prove that the general solution of tan θ = tan ∝ is given by θ = nπ +∝, n ∈ Z.

Solution:

We have,

tan θ = tan ∝

⇒ sin θ/cos θ - sin ∝/cos ∝ = 0

⇒ (sin θ cos ∝ - cos θ sin ∝)/cos θ cos ∝ = 0

⇒ sin (θ - ∝)/cos θ cos ∝ = 0

⇒ sin (θ - ∝) = 0

⇒ sin (θ - ∝) = 0

⇒ (θ - ∝) = nπ, where n ∈ Z (i.e., n = 0, ± 1, ± 2, ± 3,…….), [Since we know that the θ = nπ, n ∈ Z is the general solution of the given equation sin θ = 0]

⇒ θ = nπ + ∝, where n ∈ Z (i.e., n = 0, ± 1, ± 2, ± 3,…….)

Hence, the general solution of tan θ = tan ∝ is θ = nπ + ∝, where n ∈ Z (i.e., n = 0, ± 1, ± 2, ± 3,…….)

Note: The equation cot θ = cot ∝ is equivalent to tan θ = tan ∝ (since, cot θ = 1/tan θ and cot ∝ = 1/tan ∝). Thus, cot θ = cot ∝ and tan θ = tan ∝ have the same general solution.

Hence, the general solution of cot θ = cot ∝ is θ = nπ + ∝, where n ∈ Z (i.e., n = 0, ± 1, ± 2, ± 3,…….)

1. Solve the trigonometric equation tan θ = $\frac{1}{√3}$

Solution:

tan θ = $\frac{1}{√3}$

⇒ tan θ = tan $\frac{π}{6}$

⇒ θ = nπ + $\frac{π}{6}$, where n ∈ Z (i.e., n = 0, ± 1, ± 2, ± 3,…….), [Since, we know that the general solution of tan θ = tan ∝ is θ = nπ + ∝, where n ∈ Z (i.e., n = 0, ± 1, ± 2, ± 3,…….)]

2. What is the general solution of the trigonometric equation tan x + tan 2x + tan x tan 2x = 1?

Solution:

tan x + tan 2x + tan x tan 2x = 1

tan x + tan 2x = 1 - tan x tan 2x

$\frac{tan x  +  tan 2x}{1  -  tan x tan 2x}$ = 1

tan 3x = 1

tan 3x = tan $\frac{π}{4}$

3x = nπ + $\frac{π}{4}$, where n = 0, ± 1, ± 2, ± 3,…….

x = $\frac{nπ}{3}$ + $\frac{π}{12}$, where n = 0, ± 1, ± 2, ± 3,…….

Therefore, the general solution of the trigonometric equation tan x + tan 2x + tan x tan 2x = 1 is x = $\frac{nπ}{3}$ + $\frac{π}{12}$, where n = 0, ± 1, ± 2, ± 3,…….

3. Solve the trigonometric equation tan 2θ = √3

Solution:

tan 2θ = √3

⇒ tan 2θ = tan $\frac{π}{3}$

⇒ 2θ = nπ + $\frac{π}{3}$, where n ∈ Z (i.e., n = 0, ± 1, ± 2, ± 3,…….), [Since, we know that the general solution of tan θ = tan ∝ is θ = nπ + ∝, where n ∈ Z (i.e., n = 0, ± 1, ± 2, ± 3,…….)]

⇒ θ = $\frac{nπ}{2}$ + $\frac{π}{6}$, where n ∈ Z (i.e., n = 0, ± 1, ± 2, ± 3,…….)

Hence, the general solution of tan 2θ = √3 is θ = $\frac{nπ}{2}$ + $\frac{π}{6}$, where n ∈ Z (i.e., n = 0, ± 1, ± 2, ± 3,…….)

4. Find the general solution of the trigonometric equation 2 tan x - cot x + 1 = 0

Solution:

2 tan x - cot x + 1 = 0

⇒ 2 tan x - $\frac{1}{tan x }$ + 1 = 0

⇒ 2 tan$^{2}$ x + tan x - 1 = 0

⇒ 2 tan$^{2}$ x + 2tan x - tan x - 1 = 0

⇒ 2 tan x(tan x + 1) - 1(tan x + 1) = 0

⇒ (tan x + 1)(2 tan x - 1) = 0

⇒ either tan x + 1 = or, 2 tan x - 1 = 0

⇒ tan x = -1 or, tan x  = $\frac{1}{2}$

⇒ tan x = ($\frac{-π}{4}$) or, tan x  = tan α, where tan α = $\frac{1}{2}$

⇒ x = nπ + ($\frac{-π}{4}$) or, x = mπ + α, where tan α = $\frac{1}{2}$ and m = 0, ± 1, ± 2, ± 3,…….

⇒ x = nπ - ($\frac{π}{4}$) or, x = mπ + α, where tan α = $\frac{1}{2}$ and m = 0, ± 1, ± 2, ± 3,…….

Therefore the solution of the trigonometric equation 2 tan x - cot x + 1 = 0 are x = nπ - ($\frac{π}{4}$)  and x = mπ + α, where tan α = $\frac{1}{2}$ and m = 0, ± 1, ± 2, ± 3,…….

5. Solve the trigonometric equation tan 3θ  + 1 = 0

Solution:

tan 3θ  + 1 = 0

tan 3θ  = - 1

⇒ tan 3θ = tan (-$\frac{π}{4}$)

⇒ 3θ = nπ + (-$\frac{π}{4}$), where n ∈ Z (i.e., n = 0, ± 1, ± 2, ± 3,…….), [Since, we know that the general solution of tan θ = tan ∝ is θ = nπ + ∝, where n ∈ Z (i.e., n = 0, ± 1, ± 2, ± 3,…….)]

⇒ θ = $\frac{nπ}{3}$ - $\frac{π}{12}$, where n ∈ Z (i.e., n = 0, ± 1, ± 2, ± 3,…….)

Hence, the general solution of tan 3θ  + 1 = 0 is θ = $\frac{nπ}{3}$ - $\frac{π}{12}$, where n ∈ Z (i.e., n = 0, ± 1, ± 2, ± 3,…….)

● Trigonometric Equations

• General solution of the equation sin x = ½
• General solution of the equation cos x = 1/√2
• General solution of the equation tan x = √3
• General Solution of the Equation sin θ = 0
• General Solution of the Equation cos θ = 0
• General Solution of the Equation tan θ = 0
• General Solution of the Equation sin θ = sin ∝
  
• General Solution of the Equation sin θ = 1
• General Solution of the Equation sin θ = -1
• General Solution of the Equation cos θ = cos ∝
• General Solution of the Equation cos θ = 1
• General Solution of the Equation cos θ = -1
• General Solution of the Equation tan θ = tan ∝
• General Solution of a cos θ + b sin θ = c
• Trigonometric Equation Formula
• Trigonometric Equation using Formula
• General solution of Trigonometric Equation
• Problems on Trigonometric Equation

11 and 12 Grade Math

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