Solving Trigonometric Equations

Unlike normal solutions of algebraic equations with the number of solutions based on the degree of the variable, in trigonometric equations, the solutions are of two types, based on the different value of angle for the trigonometric function, for the same solution. For example, for a simple trigonometric equation 2Cosθ – 1 = 0, the solution is given by, Cosθ = 1/2 and, the θ values are π/3, 5π/3, 7π/3, 11π/3, and so on as the values of the cosine function repeat after every 2π radians and cos x is positive in the first and fourth quadrants. We have two types of solutions to the trigonometric equations:

• Principal Solution: The initial values of angles for the trigonometric functions are referred to as principal solutions. The solution of Sinx and Cosx repeat after an interval of 2π, and the solution of Tanx repeat after an interval of π. The solutions of these trigonometric equations, for which x lies between 0 and 2π, are called principal solutions.
• General Solution: The values of the angles for the same answer of the trigonometric function are referred to as the general solution of the trigonometric function. The solutions of trigonometric equations beyond 2π are all consolidated and expressed as a general solution of the trigonometric equations. The general solutions of Sinθ, Cosθ, Tanθ are as follows.
  • Sinθ = Sinα, and the general solution is θ = nπ + (-1)ⁿα, where n ∈ Z
  • Cosθ = Cosα, and the general solution is θ = 2nπ + α, where n ∈ Z
  • Tanθ = Tanα, and the general solution is θ = nπ + α, where n ∈ Z

Steps to Solve Trigonometric Equations

The following steps are to be followed, for solving a trigonometric equation.

• Transform the given trigonometric equation into an equation with a single trigonometric ratio (sin, cos, tan)
• Change the equation with the trigonometric equation, having multiple angles, or submultiple angles into a simple angle.
• Now represent the equation as a polynomial equation, quadratic equation, or linear equation.
• Solve the trigonometric equation similar to normal equations, and find the value of the trigonometric ratio.
• The angle of the trigonometric ratio or the value of the trigonometric ratio represents the solution of the trigonometric equation.

Examples of Solving Trigonometric Equations

Example 1: Find the principal solutions of the trigonometric equation sin x = √3/2.

Solution: To find the principal solutions of sin x = √3/2, we know that sin π/3 = √3/2 and sin (π – π/3) = √3/2

⇒ sin π/3 = sin 2π/3 = √3/2

We can find other values of x such that sin x = √3/2, but we need to find only those values of x such that x lies in [0, 2π] because a principal solution lies between 0 and 2π.

So, the principal solutions of sin x = √3/2 are x = π/3 and 2π/3.

Example 2: Find the solution of cos x = 1/2.

Solution: In this case, we will find the general solution of cos x = 1/2. We know that cos π/3 = 1/2, so we have

cos x = 1/2

⇒ cos x = cos π/3

⇒ x = 2nπ + (π/3), where n ∈ Z —- [Using Cosθ = Cosα, and the general solution is θ = 2nπ + α, where n ∈ Z]

Therefore, the general solution of cos x = 1/2 is x = 2nπ + (π/3), where n ∈ Z.

Important Notes on Trigonometric Equations

• For any real numbers x and y, sin x = sin y implies x = nπ + (-1)ⁿy, where n ∈ Z.
• For any real numbers x and y, cos x = cos y implies x = 2nπ ± y, where n ∈ Z.
• If x and y are not odd multiples of π/2, then tan x = tan y implies x = nπ + y, where n ∈ Z.
• sin A = 0 implies A = nπ and cos A = 0 implies A = (2n + 1)π/2, where n ∈ Z