+2kπ, where k is any integer. Solution: 7. ), 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,……. Hence for such equations, we have to find the values of x or find the solution. Theorem 1: For any real numbers x and y, sin x = sin y implies x = nπ + (–1)n y, where n ∈ Z. ∈ 2, ± 3,……. Find the general solution of the trigonometric equation 2 tan x - cot x + 1 = 0, ⇒ 2 tan x - \(\frac{1}{tan x }\) + 1 = 0, ⇒ 2 tan\(^{2}\) x + 2tan x - tan x - 1 = 0, ⇒ 2 tan x(tan x + 1) - 1(tan x + 1) = 0, ⇒ 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 = \(\frac{1}{2}\). See the attachment and follow the procedure . These equations have one or more trigonometric ratios of unknown angles. Where E1 and E2 are rational functions. The set of all solutions to sin(x) = y is. and x = −sin-1(y) + (2k+1)π. where k can be any integer; that is, the solutions for x Also, the character ‘Z’ is used to denote the set of integers. ⇒ x = nπ - (\(\frac{π}{4}\)) or, x = mπ + α, where tan α = \(\frac{1}{2}\) and m = 0, ± 1, ± 2, ± 3,……. Case 2: -1 > y or y > 1 , Index | Geometrical Interpretation y, Note In case, two angles one with positive sign and the other with negative sign qualify for the numerically least angle, then it is the convention to select the angle with positive sign as principal value. © and ™ math-only-math.com. Or want to know more information SSL Certificate | What is an SSL Certificate? 3x = nπ + \(\frac{π}{4}\), where n = 0, ± 1, ± 2, ± 3,……. Z But, we know that if sin x = 0, then x = 0, π, 2π, π, -2π, -6π, etc. Z (i.e., n = 0, ± 1, ± 2, ± 3,……. A trigonometric equation will also have a general solution expressing all the values which would satisfy the given equation, and it is expressed in a generalized form in terms of ‘n’. Table 2 Let us find the general solutions of  2cos x = 1. A visual aide illustrating the derivation of the general solution to the equation: tan(x) = k (where k is a real number) Use this Google Search to find what you need. sin (x – y) = 0     [By trigonometric identity]. Now . Solution: 5. For principal value, we have to take numerically smallest angle. ), Hence, the general solution of tan θ = tan ∝ is θ = nπ + ∝, where n ∈ To learn more about trigonometric equations, trigonometry, please download BYJU’S- The Learning App. For How To Get SSL Certificate? For example, cos 2 x + 5 sin x = 0 is a trigonometric equation. Since sin θ is negative, θ will be in 3rd or 4th quadrant. 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,……. x = π/4 +2kπ or x = 3π/4 ), If we take anticlockwise direction the numerical value of the angle will be greater than π. (1) Roots of trigonometrical equation: The value of unknown angle (a variable quantity) which satisfies the given equation is called the root of an equation, e.g., cos θ = ½, the root is θ = 60° or θ = 300° because the equation is satisfied if we put θ = 60° or θ = 300°. 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