# Trigonometric Equations

Trigonometric Equations are mathematical equations made up of expressions that contain trigonometric functions such as sine, cosine, and tangent. These equations create relationships between angles and sides of triangles or we can say Trigonometric Equations represent various relationships between trigonometric functions.

Trigonometric Equations help us find values for the angles or sides that meet the specified criteria, and these angles are the solution of Trigonometric Equations. The measure of these angles in Radians or degrees can be used to express solutions.

Trigonometric Equations require the use of trigonometric identities and specific angles and are used in many professions, including physics, engineering, astronomy, and architecture, where a thorough grasp of angles and their connections is essential for practical calculations and problem-solving. This article will help you learn about these equations i.e., Trigonometric Equations.

## What are Trigonometric Equations?

Trigonometric functions of angles are used as variables in trigonometric equations. In trigonometric equations, the angle of Î¸ trigonometric functions such as sin Î¸, cos Î¸, and tan Î¸ is employed as a variable. Trigonometric equations, like ordinary polynomial equations, have solutions that are referred to as principle solutions and general solutions.

To solve the trigonometric equations, we will use the information that the period of sin x and cos x is 2Ï€ and the period of tan x is. Let us learn more about trigonometric equations, how to solve them, and how to identify their solutions using a few solved examples of trigonometric equations.

### Trigonometric Equations Examples

As Trigonometric Equations represent the relationships between different trigonometric functions, there can be infinitely many Trigonometric Equations. Some examples of Trigonometric Equations are:

• sin(x) = 1/âˆš2
• cos(3x) = -1/2
• 2sin(2x) – 1 = 0
• tan(2x) + 3 = 0
• 2 cos(x) + sin(2x) = 1
• 3 sin(x) – 2 cos(2x) = 1
• 2 sin(3x) +  tan(x) = 0
• cot(x) + 2 csc(x) = 0
• 4 cos(2x) – 3 sin(3x) = 2

## How to Solve Trigonometric Equations?

To solve a trigonometric equation, use the procedures below.

• Transform the supplied trigonometric equation into a single trigonometric ratio equation (sin, cos, tan).
• Convert the equation with many angles or submultiple angles into a simple angle using the trigonometric equation.
• Now, write the equation as a polynomial, quadratic, or linear equation.
• Solve the trigonometric problem in the same way you would any other equation, then calculate the trigonometric ratio.
• The solution of the trigonometric equation is represented by the angle of the trigonometric ratio or by the value of the trigonometric ratio.

## General Solutions Trigonometric Equations

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

### General Solutions

sin Î¸ = 0 Î¸ = nÏ€
cos Î¸ = 0 Î¸ = (nÏ€ + Ï€/2)
tan Î¸ = 0 Î¸ = nÏ€
sin Î¸ = 1 Î¸ = (2nÏ€ + Ï€/2) = (4n+1)Ï€/2
cos Î¸ = 1 Î¸ = 2nÏ€
sin Î¸ = sin Î±Î¸ = nÏ€ + (-1)nÎ±, Where Î± âˆˆ [-Ï€/2, Ï€/2]
cos Î¸ = cos Î±Î¸ = 2nÏ€ Â± Î±, Where Î± âˆˆ (0, Ï€]
tan Î¸ = tan Î±Î¸ = nÏ€ + Î±, Where Î± âˆˆ (-Ï€/2 , Ï€/2]
sin 2Î¸ = sin 2Î±Î¸ = nÏ€ Â± Î±
cos 2Î¸ = cos 2Î±Î¸ = nÏ€ Â± Î±
tan 2Î¸ = tan 2Î±Î¸ = nÏ€ Â± Î±

If Î± is supposed to be the least positive number that satisfies two specified trigonometrical equations, then the general value of Î¸ will be 2nÏ€ + Î±.

### Principle Solution of Trigonometric Equations

The principal solution of a trigonometric equation refers to the solution that falls within a specific interval, typically between 0Â° and 360Â° or 0 and 2Ï€ 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 = nÏ€ + (â€“1)ny, where n âˆˆ Z
• cos x = cos y, which implies x = 2nÏ€ Â± y, where n âˆˆ Z
• tan x = tan y implies x = nÏ€ + y, where n âˆˆ Z

Let’s discuss these theorems in detail.

### Theorem 1: If x and y are real integers, sin x = sin y implies x = nÏ€ + (â€“1)ny, where n âˆˆ Z

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

sin x = sin y

â‡’ sin x – sin y = 0

â‡’ sin x – sin y = 0

â‡’ 2cos (x + y)/2 sin (x – y)/2 = 0

â‡’ cos (x + y)/2 = 0 or sin (x – y)/2 = 0

Taking the common answer from both requirements, we obtain:

x = nÏ€ + (-1)ny, where n âˆˆ Z

### Theorem 2: For any two real integers x and y, cos x = cos y, which implies x = 2nÏ€ Â± y, where n âˆˆ Z.

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

cos x – cos y = 0.

â‡’ 2sin (x + y)/2 sin (y – x)/2 = 0

â‡’ sin (x + y)/2 = 0 or sin (x – y)/2 = 0

â‡’ (x + y)/2 = nÏ€ or (x â€“ y)/2 = nÏ€

Taking the common answer from both criteria yields:

x = 2nÏ€Â± y, where n âˆˆ Z

### Theorem 3: Show that tan x = tan y implies x = nÏ€ + y, where n âˆˆ Z if x and y are not odd multiples of Ï€/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

â‡’ sin x cos y – sin y cos x

â‡’ sin x cos y – sin y cos x = 0

â‡’ sin (x – y) = 0

As a result, x – y =nÏ€ or x = nÏ€ + y, where n âˆˆ Z.

## Formulas of Trigonometric Equations

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 = nÏ€ + (-1)n y, where n âˆˆ Z.
• For any two real integers, x and y, Cos x = cos y implies x = 2nÏ€ Â± y, where n âˆˆ Z.
• If x and y are not odd multiples of Ï€/2, then tan x = tan y implies that x = nÏ€ + y, where n âˆˆ Z.

## Solving Trigonometric Equations

Example 1: Determine the primary solution to the trigonometric equation tan x = -âˆš3

Solution:

We have tan x = -âˆš3 here, and we know that tan /3 = âˆš3. So there you have it.

tan x = -âˆš3

â‡’ tan x = – tan Ï€/3

â‡’ tan x = tan(Ï€ – Ï€/3) Alternatively, tan x = tan(2Ï€ – Ï€/3)

â‡’ tan x = tan 2Ï€/3 OR tan x = tan 5/3.

As a result, the primary solutions of tan x = -âˆš3 are 2Ï€/3 and 5Ï€/3

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

Example 2: Find sin 2x â€“ sin 4x + sin 6x = 0

Solution:

Given: sin 2x – sin 4x + sin 6x = 0.

â‡’sin 2x + sin 6x â€“ sin 4x = 0

â‡’2sin 4x.cos 2x â€“ sin 4x = 0

â‡’sin 4x (2cos 2x â€“ 1) = 0

â‡’sin 4x = 0 or cos 2x = 1/2

â‡’4x = nÏ€ or 2x = 2nÏ€ Â± Ï€/3

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

â‡’x = nÏ€/4 or nÏ€ Â± Ï€/6

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

Solution:

sin Ï€/6 = 1/2

sin 5Ï€/6 = sin (Ï€ – Ï€/6)

= sin Ï€/6 = 1/2

As a result, the primary answers are x =Ï€/6 and x = 5Ï€/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 Ï€/3 = 1/2, we have

cos x = 1/2

cos x = cos Ï€/3

x = 2nÏ€ + (Ï€/3), where n âˆˆ Z —- [With CosÎ¸ = CosÎ±, the generic solution is Î¸ = 2nÏ€ + Î±, where n âˆˆ Z]

As a result, cos x = 1/2 has a generic solution of x = 2nÏ€ + (Ï€/3), where n âˆˆ Z.

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

Solution:

To obtain the primary solutions of sin x = âˆš3/2, we know that sin Ï€/3 = âˆš3/2 and sin (Ï€ – Ï€/3) = âˆš3/2

sin Ï€/3 = sin 2Ï€/3 = âˆš3/2

We can discover additional values of x such that sin x = âˆš3/2, but we only need to find those values of x where x is between [0, 2Ï€] since a primary solution is between 0 and 2Ï€.

As a result, the primary solutions of sin x = âˆš3/2 are x = Ï€/3 and 2Ï€/3.

## Solve Trigonometric Equations

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) = -âˆš3

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

Problem 6: Solve for x in the equation: cot(x) = 1

Problem 7: Determine all solutions for x in the equation: 3 sin(x) + 2 cos(x) = 0

Problem 8: Find the values of x that satisfy the equation: tan(2x) = 1

Problem 9: Solve for x in the equation: sec(x) = -2

Problem 10: Find all solutions for x in the equation: 4 sin(3x) = 1

## Trigonometric Equations – FAQs

### 1. Define Trigonometric Equations.

Trigonometric equations are similar to algebraic equations in that they might be linear, quadratic, or polynomial equations. In trigonometric equations, the trigonometric ratios SinÎ¸, CosÎ¸, TanÎ¸ are used to denote the variables.

### 2. Give Some Examples of Trigonometric Equations.

The following are some instances of trigonometric equations:

• 2 Cos2x + 3 Sinx = 0
• Cos4x = Cos2x
• Sin2x – Sin4x + Sin6x = 0

### 3. Which three trigonometric equations are there?

SinÎ¸, CosÎ¸, and TanÎ¸ are the three major functions in trigonometry.

### 4. What is Sin Inverse?

The arcsin function is the inverse of the sin function. Sin, on the other hand, will not be invertible since it is not injective, and so it is not mixed (invertible). Furthermore, in order to obtain the arcsine function, the domain of sine must be limited to [âˆ’Ï€/2,Ï€/2].

### 7. How to Find Solutions of Trigonometric Equations?

To find solutions to trigonometric equations, use identities and algebraic techniques to isolate the variable. Then, apply inverse trig functions and consider the unit circle.

### 8. How do I solve Trigonometric Equations in Class 11 Math?

To solve trigonometric equations in Class 11 math:

• Isolate the trigonometric function on one side.
• Apply trigonometric identities and simplify the equation.
• Solve for the variable using inverse trigonometric functions.
• Check for extraneous solutions.
• State the solution with appropriate constraints.

### 9. What is the Principal Solution of a Trigonometric Equation?

The principal solution of a trigonometric equation is the solution within the principal interval, typically in radians: [-Ï€, Ï€] for cosine, or [0, 2Ï€] for the sine and tangent function.

### 10. What Are Trigonometric Identities, and How Are They Used in Solving Equations?

Trigonometric identities are equations involving trig functions, and these are used to simplify and solve trigonometric equations by manipulating expressions.

### 11. How Do I Find All Solutions of a Trigonometric Equation?

To find all solutions of a trigonometric equation, you can first find the principal solution and then use the periodicity of trigonometric functions to add integer multiples of the period.

Previous
Next