The trigonometric equations involve trigonometric functions of 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 use the fact that the period of sin x and cos x is 2π and the period of tan x is π to find the solutions of the trigonometric equations. Let us learn more about trigonometric equations, the method to solve them, and find their solutions with the help of a few solved examples of trigonometric equations for a better understanding of the concept.
What are Trigonometric Equations?
The trigonometric equations are similar to algebraic equations and can be linear equations, quadratic equations, or polynomial equations. In trigonometric equations, the trigonometric ratios of Sinθ, Cosθ, Tanθ are represented in place of the variables, as in a normal polynomial equation. The trigonometric ratios used in trigonometric equations are Sinθ, Cosθ, or Tanθ.
The linear equation ax + b = 0 can be written as a trigonometry equation as aSinθ + b = 0, which is also sometimes 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 unlike normal solutions of equations with the number of solutions based on the degree of the variable, in trigonometric equations, the same value of solution exists for different values of θ. For example, we have Sinθ = 1/2 = Sinπ/6 = Sin5π/6 = Sin13π/6, and so on as the values of the sine function repeat after every 2π radians.
Some of the examples of trigonometric equations are as follows.
- Sin2x – Sin4x + Sin6x = 0
- 2Cos2x + 3Sinx = 0
- Cos4x = Cos2x
- Sin2x + Cosx = 0
- Sec22x = 1 – Tan2x
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