General Solution of Some Standard Equations

Trigonometric equations are equations that use trigonometric terms in the equation. A trigonometric equation is applied to angles. The variables are the values of angles, which can be in degree or radian. The solution of a trigonometric equation is the value of the angle that satisfies the equation. The general solution of a trigonometric equation is all the values of the angle that will satisfy the equation. As trigonometric functions are periodic in nature, the general solutions of such equations can have multiple solutions. 

The trigonometric functions that exist in trigonometric equations are sin, cos, tan, sec, cosec, and cot.

For example, sin=1 is a trigonometric equation. The general solution of the given equation is (2n+/2), where n is the integer. The general solution predicts that the value of sin will be 1 at -/2,/2,5/2, and so on.

Some Equations and their General Solutions

• The equation is sin θ = 0. 

The given equation says that the value of sin at the angle is 0. Now, sine is a periodic function with its period 2. Moreover, the value of sine is 0 at angle 0. The value of sine will now repeat itself after every 2 interval. Therefore, the solution to the given equation will be θ = nπ, where n is an integer.

• The equation is cos θ = 0. 

The given equation says that the value of cos at the angle is 0. Now, cosine is a periodic function with its period 2. Moreover, the value of cosine is 0 at angle /2. The value of cosine will now repeat itself after every 2 interval. Therefore, the solution to the given equation will be θ = (nπ + π/2), where n is an integer.

• The equation is sin θ = 1. 

The given equation says that the value of sin at the angle is 1. Now, sine is a periodic function with its period 2, and the value of sine is 1 at angle /2. Moreover, the value of sine will repeat itself after every 2 interval. Therefore, the solution to the given equation will be θ = (2nπ + π/2) or (4n+1) π/2, where n is an integer.

• The equation is cos θ = 1. 

The given equation says that the value of cos at the angle is 1. Now, cosine is a periodic function with its period 2, and the value of cosine is 1 at angle 0. The value of cosine will now repeat itself after every 2 interval. Therefore, the solution to the given equation will be θ = 2nπ, where n is an integer.

• The equation is sin θ = sin α. 

The given equation says that the value of sin at the angle θ is the equation to the value of sin at α.  Now, sine is a periodic function with its period 2. So, the value of sine will repeat itself after every 2 interval. Therefore, the solution to the given equation will be θ = nπ +(-1)n α, where α ∈ [-π/2, π/2].

• The equation is cos θ = cos α. 

The given equation says that the value of cos at the angle θ is the equation to the value of cos at α. Now, cosine is a periodic function with its period 2. So, the value of sine will repeat itself after every 2 interval. Therefore, the solution to the given equation will be θ = 2nπ ± α, where α ∈ (0, π]. Remember that 0 is not included as it is in the open interval.

• The equation is tan θ = tan α. 

The given equation says that the value of tan at the angle θ is the equation to the value of tan at α. Now, the tangent is a periodic function with its period . So, the value of tan will repeat itself after every interval. Therefore, the solution to the given equation will be θ = nπ + α, where α ∈ (-π/2, π/2]. Remember that -π/2 is not included as it is in the open interval.

• The equation is tan θ = 1. 

The given equation says that the value of tan at the angle is 1. Now, the tangent is a periodic function with its period , and the value of tan is 1 at angle /4. The value of sine will now repeat itself after every interval. Therefore, the solution to the given equation will be θ = (nπ + π/4) or (4n+1) π/4, where n is an integer.

Examples

• What is the general solution to the equation tan x = 1/(√3)?

Explanation: The given equation is tan x = 1/(√3).

We know that tan /6 = 1/(√3)

So, the general solution to the given equation will be x = (nπ + π/6) or (4n+1) π/6, where n is an integer.

• What are the general solutions to the equation sin x =1/2.

Explanation: The given equation is sin x =1/2.

It is known that sin π/6 = 1/2.

Therefore, the general solutions to the equation are x = (2nπ + π/6) or (12n+1) π/6.

Conclusion

Trigonometric equations are equations including trigonometric functions and the variable is the angle. The general solutions to the equations are all the values of the angle that satisfy the equation. General solutions will be multiple, as trigonometric functions are periodic.