Important Topic of Maths - Trigonometric Equation

The trigonometric equations include trigonometric functions in which angles as variables. The angle of θ trigonometric functions such as Sinθ, Cosθ, Tanθ is used as a variable in trigonometric equations. Similar to general polynomial equations, the trigonometric equations also have solutions, which are referred to as principal solutions, and general solutions. We will discus here types of solving equation, trigonometric formulas and steps of trigonometric equation.

Trigonometric equations

The trigonometric equations are similar to algebraic equation and can be linear equations, quadratic equations, or polynomial equations. But in the case of trigonometric equations, the trigonowhametric ratios of Sinθ, Cosθ, Tanθ are represented in place of the variables, as in a normal polynomial equation.

Trigonometric equation can be write as Sinθ + b = 0, can also be written as Sinθ = Sinα. The quadratic equation ax² + bx + c = 0 is as an example of trigonometric equation is written as aCos²θ + bCosθ + c = 0. But solution of trigonometric equation dependent angle of trigonometric ratios. For example, we have Sinθ = 1/2 = Sinπ/6 = Sin5π/6 = Sin13π/6, and so on as the values of the every trigonometric ratio repeat after every 2π radians.

Also Check - Hybridization Of Ethene

Trigonometric Equation Formulas

We have some results and general solutions for solving other trigonometric equations. These results are as follows:

• 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.

More About - Hybridization of Methane

Types of the solution of the trigonometric Equation

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 Sin x and Cos x repeat after an 2π radian, and the solution of Tan x repeat after an interval of π. The solutions for every trigonometric equations, for which x lies between 0 and 2π, are called principal solutions.
• General Solution: If values of the angles for the same answer of the trigonometric function gives to 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 for these 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

Step to solve Trigonometric Equation

Solving the trigonometric equation some steps are as follow.

• Convert the given trigonometric equation into an equation with a single trigonometric ratio (sin, cos, tan)
• Change the multiple angles, or submultiple angles of trigonometric equation 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 shows the solution of the trigonometric equation.