Trigonometric Equations

Trigonometric equations are mathematical expressions that involve trigonometric functions (such as sine, cosine, tangent, etc.) and are set equal to a value.

For example, a simple trigonometric equation might be: sin⁡(x) = 0.5

Solving this equation involves finding the values of x that make the sine of x equal to 0.5. The solutions could be periodic due to the nature of trigonometric functions.

Note: To solve the trigonometric equations, we will use the information that the period of sin x and cos x is 2π and the period of tan x is π.

We usually solve trigonometric equations to find the value(s) of the angle (like x or θ) that make the equation true.

For example: sin x = 1/2

Solutions: x = 30° + 360°n OR x = 150° + 360°n (where n is any integer)

How to Solve Trigonometric Equations?

The following steps define how to solve trigonometric equations:

• Step 1: Transform the supplied trigonometric equation into a single trigonometric ratio equation (sin, cos, tan).
• Step 2: Convert the equation with many angles or submultiple angles into a simple angle using the trigonometric equation.
• Step 3: Now, write the equation as a polynomial, quadratic, or linear equation.
• Step 4: Solve the trigonometric problem in the same way you would any other equation, then calculate the trigonometric ratio.
• Step 5: The solution of a trigonometric equation is given by the angle(s) (in degrees or radians) that satisfy the equation.

General Solutions Trigonometric Equations

The table below lists the generic solutions to the trigonometric functions defined in equations.

If α is supposed to be the least positive number that satisfies two specified trigonometrical equations, then the general value of θ will be 2nπ + α.

Principle Solution of Trigonometric Equations

The principal solution of a trigonometric equation refers to the solution that falls within a specific interval, typically between 0° and 360° or 0 and 2π radians. This solution represents the primary or fundamental solution of the equation, and it is often used as a reference point when finding other solutions.

Proof of Solutions of Trigonometric Equations

Let us now use theorems to demonstrate these solutions i.e.,

• sin x = sin y implies x = nπ + (–1)ⁿy, where n ∈ Z
• cos x = cos y, which implies x = 2nπ ± y, where n ∈ Z
• tan x = tan y implies x = nπ + y, where n ∈ Z

Theorem 1: If x and y are real numbers, sin x = sin y implies x = nπ + (–1)ⁿy, where n ∈ Z

Proof:

Consider the following equation: sin x = sin y. Let's try to solve this trigonometric equation in general.

sin x = sin y 
⇒ sin x - sin y = 0 
⇒ sin x - sin y = 0 
⇒ 2cos (x + y)/2 sin (x - y)/2 = 0 
⇒ cos (x + y)/2 = 0 or sin (x - y)/2 = 0

Taking the common answer from both requirements, we obtain:

x = nπ + (-1)ny, where n ∈ Z

Theorem 2: For any two real integers x and y, cos x = cos y, which implies x = 2nπ ± y, where n ∈ Z.

Proof:

Likewise, the generic solution of cos x = cos y is: 

cos x - cos y = 0.
⇒ 2sin (x + y)/2 sin (y - x)/2 = 0
⇒ sin (x + y)/2 = 0 or sin (x - y)/2 = 0
⇒ (x + y)/2 = nπ or (x – y)/2 = nπ

Taking the common answer from both criteria yields:

x = 2nπ± y, where n ∈ Z

Theorem 3: Show that tan x = tan y implies x = nπ + y, where n ∈ Z if x and y are not odd multiples of π/2.

Proof:

Similarly, we may utilise the conversion of trigonometric equations to obtain the solution to equations involving tan x or other functions.

In other words, if tan x = tan y,

Then, sin x cos x = sin y cos y 
⇒ sin x cos y - sin y cos x 
⇒ sin x cos y - sin y cos x = 0 
⇒ sin (x - y) = 0

As a result, x - y = nπ or x = nπ + y, where n ∈ Z.

Trigonometric Equations Formulas

For solving other trigonometric equations, we use some of the conclusions and general solutions of the fundamental trigonometric equations. The following are the outcomes:

• For any two real integers, x and y, sin x = sin y means that x = nπ + (-1)ⁿ y, where n ∈ Z.
• For any two real integers, 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 that x = nπ + y, where n ∈ Z.

Solved Examples

Example 1: Determine the primary solution to the trigonometric equation tan x = -√3

Solution:

We have tan x = -√3 here, and we know that tan /3 = √3. So there you have it.

tan x = -√3

⇒ tan x = - tan π/3

⇒ tan x = tan(π - π/3) Alternatively, tan x = tan(2π - π/3)

⇒ tan x = tan 2π/3 OR tan x = tan 5/3.

As a result, the primary solutions of tan x = -√3 are 2π/3 and 5π/3

The primary answers are x = 2π/3 and x = 5π/3.

Example 2: Find sin 2x – sin 4x + sin 6x = 0

Solution:

Given: sin 2x - sin 4x + sin 6x = 0.

⇒ sin 2x + sin 6x – sin 4x = 0
⇒ 2sin 4x.cos 2x – sin 4x = 0
⇒ sin 4x (2cos 2x – 1) = 0
⇒ sin 4x = 0 or cos 2x = 1/2
⇒ 4x = nπ or 2x = 2nπ ± π/3

As a result, the general solution to the above trigonometric problem is as follows:

⇒ x = nπ/4 or nπ ± π/6

Example 3: Determine the primary solution to the equation sin x = 1/2.

Solution:

We already know that

sin π/6 = 1/2

sin 5π/6 = sin (π - π/6)

= sin π/6 = 1/2

As a result, the primary answers are x =π/6 and x = 5π/6.

Example 4: Determine the answer to cos x = 1/2.

Solution:

In this example, we'll use the general solution of cos x = 1/2. Because we know that cos π/3 = 1/2, we have 

cos x = 1/2 

cos x = cos π/3 

x = 2nπ + (π/3), where n ∈ Z ---- [With Cosθ = Cosα, the generic solution is θ = 2nπ + α, where n ∈ Z]

As a result, cos x = 1/2 has a generic solution of x = 2nπ + (π/3), where n ∈ Z.

Example 5: Determine the primary solutions to the trigonometric equation sin x = √3/2.

Solution:

To obtain the primary solutions of sin x = √3/2, we know that sin π/3 = √3/2 and sin (π - π/3) = √3/2

sin π/3 = sin 2π/3 = √3/2

We can discover additional values of x such that sin x = √3/2, but we only need to find those values of x where x is between [0, 2π] since a primary solution is between 0 and 2π.

As a result, the primary solutions of sin x = √3/2 are x = π/3 and 2π/3.

Practice Questions

Problem 1: Solve for x in the equation: sin(x) = 1/2

Problem 2: Find all solutions for x in the equation: 2 cos(2x) = 1

Problem 3: Determine the solutions for x in the equation: tan(x) = -√3

Problem 4: Solve for x in the equation: 3 sin(x) - 4 cos(x) = 0

Problem 5: Find the solutions for x in the equation: 2 sin(2x) + 1 = 0