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Trigonometric equations are the equations with the involvement of one or more trigonometric ratios of variable angles. The angles used in the equation will be the variables. Generally, the value of angle is taken in radian. The trigonometric equations follow a certain condition which is dependent on the value of the trigonometric ratio. The trigonometric ratios or functions that are involved in the trigonometric equation are sin, cos, tan, cosec, sec, and cot.
For example, sin=x is a trigonometric equation. In the given equation, the value of can be any real number but the value of x can vary between -1 and +1. This is because the value of sine trigonometric ratio or function can vary between -1 and +1 only.
A trigonometric equation is satisfied for a certain value of angle. These values of angles are known as its solution. Some of the trigonometric equations are satisfied by any real value of the variable or angle, and hence they are called trigonometric identities. The trigonometric equations, which are true for only a few values of angle, are normally called conditional equations.
Some trigonometric equations
Consider a trigonometric equation sin x = 1.
The given equation estimates the value of sin at the angle x as 1. The equation will be true for all the values of x such that sin x is equal to 1. We know that sin 90 is equal to 1. So, the possible solution of the equation will be 90 degrees or /2 radian. Sine is a periodic function with period 2. So, the general solution of the equation will be x = (2nπ + π/2) or (4n+1) π/2 where n is any integer.
Consider a trigonometric equation cos x = 1.
The given equation estimates the value of cos at the angle x as 1. The equation will be true for all the values of x such that cos x is equal to 1. It is known that cos 0 is equal to 1. So, the possible solution of the equation will be 0 degree or radian. Cosine is a periodic function with period 2. So, the general solution of the equation will be x = 2nπ where n is an integer.
Let’s consider the equation sin x = sin y.
The given equation equates the value of sin at the angle x and the value of sin at y. It is required to find the solution of the given equation in terms of x and y. Now, sine is a periodic function with its period 2. So, it’s value will repeat after every 2 interval. Therefore, the solution of the given equation can be written as x = nπ +(-1)n y, where y ∈ [-π/2, π/2] and n is an integer.
Consider the equation cos x = cos y.
The given equation equates the value of cos at the angle x and the value of cos at y. It is required to find the solution of the given equation in terms of x and y. Cosine is a periodic function with its period 2. So, it’s value will repeat after every 2 interval. Therefore, the solution of the given equation will be x = 2nπ ± y, where y ∈ (0, π] and n is an integer. Remember that 0 is not included because it is in the open interval.
Consider the equation tan x = tan y.
The given equation equates the value of tan at the angle x and the value of tan at y. The question requires the solution of the equation tan x = tan y is the value of x in terms of y. Tangent function (or tan) is a periodic function with a period of . So, the value of Tangent will repeat after every interval. Therefore, the solution of the given equation can be represented as x = nπ + y, where y ∈ (-π/2, π/2] and n is an integer. Remember that -π/2 is not included as it is in the open interval.
The solution of the above equations is the general solution.
Examples
Check whether the equation tan2 x+tanx+4 = 0 has any real solution or not.
Explanation: Given equation is tan2 x+tanx+4 = 0.
We know that the given equation is a quadratic equation in terms of tan (trigonometric ratio). To check the nature of the solution we need to find the roots of the equation.
The discriminant of the given equation will be,
D=12-4412=-72
The discriminant of the given equation is negative. So, the roots of the equation will not be real.
Thus, the solution of the given equation will have complex value (No real value).
Determine the solutions of the equation cos x =1/2.
Explanation: The given equation is cos x =1/2.
It is known that, cos π/3 = 1/2.
Therefore, the solutions of the equation are x = (2nπ + π/3) or (6n+1) π/3.
Remember that the solution of a trigonometric equation can be found easily for some specific angles only. For some values of the ratio, the solution is very difficult to find and hence we can use a calculator for the same.
Conclusion
Trigonometric equations are equations that involve trigonometric functions or ratios. The variable of the equation is the value of the angle. The solution of any trigonometric equation represents the value of the variables which satisfy the equation. There are two types of trigonometric equations in general; trigonometric identities and conditional equations. Trigonometric identities are true for all the values of a variable, and the conditional equations are true only for some specific values of the variable. The trigonometric ratios which are involved in any trigonometric equation are sin, cos, tan, sec, cosec, and cot.
