Solutions of Simple Trigonometric Equations

Equations in trigonometry use trigonometric functions with angles as variables, thus these equations require trigonometric functions. In trigonometric equations, the angle of trigonometric functions such as Sin, Cos, and Tan is treated as a variable and employed in the calculation. Trigonometric equations, just like generic polynomial equations, have solutions, which can be broken down into two categories: primary solutions and general solutions. Principal solutions are the most straightforward answers, while general solutions are the most comprehensive.

To solve the trigonometric equations, we will make use of the fact that the period of sin x and cos x is 2 and that the period of tan x is. This information will allow us to determine the solutions to the equations. In order to gain a deeper comprehension of the idea at hand, let us continue our study of trigonometric equations, the method used to solve them, and the solutions to these equations with the assistance of a few previously solved examples of trigonometric equations.

What are Trigonometric Equations?

Equations in trigonometry can take the form of linear equations, quadratic equations, or polynomial equations, just like algebraic equations. They are quite similar to algebraic equations. In trigonometric equations, the trigonometric ratios of Sin, Cos, and Tan are represented in place of the variables, similar to how it is done in a conventional polynomial equation. Sin, Cos, and Tan are the trigonometric ratios that are utilised whenever a trigonometric equation is being solved.

The linear equation ax + b = 0 can also be expressed as a trigonometric equation as aSinθ + b = 0, which is also commonly written as Sinθ = Sinα. Both of these forms of the equation are equivalent to each other. The trigonometric equation for the quadratic equation ax² + bx + c = 0 is expressed as aCos²θ + bCosθ + c = 0. Another example of a trigonometric equation is the quadratic equation. However, in contrast to typical solutions of equations, this one has an infinite number of solutions. In trigonometric equations, the same value of solution occurs for different values of. This is because the degree of the variable determines whether or not the solution exists. For instance, we get  Sinθ = 1/2 = Sinπ/6 = Sin5π/6 = Sin13π/6, and so on since the values of the sine function repeat after every 2π  radians.

The following are some instances of trigonometric equations that you can come across.

• Sin2x – Sin4x + Sin6x = 0

• 2Cos²x + 3Sinx = 0

• Cos4x = Cos2x

• Sin2x + Cosx = 0

• Sec²2x = 1 – Tan2x

Trigonometric Equations Formulas

In order to solve other trigonometric equations, we make use of the results and general solutions that we obtained from solving the basic trigonometric equations. The following is what these findings are:

• If sin x = sin y, then x must equal x = nπ + (-1)ⁿy, where n ∈ Z.

• If cos x = cos y for any real numbers x and y, then x = 2nπ ± y, where n ∈ Z.

• If x and y are not odd multiples of /2, then the equation tan x = tan y implies that x = nπ + y, where n ∈ Z.

Solving Trigonometric Equations

Unlike regular algebraic equations, which have a fixed number of solutions depending on the variable’s degree, trigonometric equations have two types of solutions, each based on a distinct value of angle for the trigonometric function. For example, the solution to the simple trigonometric equation Cosθ = 1/2 is  Cosθ = 1/2 , and the values are π/3, 5π/3, 7π/3, 11π/3, and so on, because the cosine function values repeat every 2 radians and cos x is positive in the first and fourth quadrants. The trigonometric equations have two types of solutions:

• Principal Solution: For trigonometric functions, the initial values of angles are referred to as principal solutions. Sinx and Cosx solutions repeat after a 2π interval, while Tanx solutions repeat after a π. Principal solutions are the answers to these trigonometric equations where x is between 0 and 2.

• The generic solution of the trigonometric function is defined as the values of the angles given the same answer of the trigonometric function. All trigonometric equation answers beyond 2 are collected and given as a general trigonometric equation solution. The following are the general solutions for Sinθ, Cosθ, Tanθ .

• Sinθ = Sinα,The generic answer 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 answer is θ = nπ + α, where n ∈ Z.

Conclusion

Equations in trigonometry can take the form of linear equations, quadratic equations, or polynomial equations, just like algebraic equations. They are quite similar to algebraic equations. In trigonometric equations, the trigonometric ratios of Sin, Cos, and Tan are represented in place of the variables, similar to how it is done in a conventional polynomial equation. In normal solutions of algebraic equations, the number of solutions is based on the degree of the variable. However, in trigonometric equations, the solutions are of two different types, based on the different values of angle for the trigonometric function. This is in contrast to the number of solutions being based on the degree of the variable in normal solutions of algebraic equations.