Table of Contents
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, 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 trigonometric 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.
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.
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.
FAQs
Q. Give the solution of the cos x = 1/2
Ans: 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.
How many types of the solution?
there are two types of solutions. 1. Principal solution 2. General solution
What is trigonometric Equation?
The trigonometric equations are like as to solving the algebraic equations it also be like linear equations, quadratic equations, or polynomial equations. In this, the trigonometric ratios of Sinθ, Cosθ, Tanθ are similar as the variables.
