Trigonometric Equations: General Solutions, Formula, Proof & Examples

Trigonometric Equation is an equation that has either one or more trigonometric ratios of undefined angles. Trigonometric Equations are expressed by the functions, namely sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (cosec) angles. 

• Trigonometry expresses the relationship between the length and angles of the sides of triangles. 
• It is derived from a combination of two Greek words, namely trigonon (triangle) and metron (measure).
• For instance, cos²x + sin²x = 1 is a trigonometric equation. 
• All possible solutions of a trigonometric equation are known as principal solutions or general solutions.
• It is an equation involving one or more trigonometric functions with a variable.
• Trigonometric equations can be used to find unknown angles or sides of a right-angled triangle.

Read More: NCERT Solutions for Class 11 Mathematics Trigonometric Functions

Key Terms: Trigonometry, Trigonometric Equations, Trigonometric Values, Trigonometric Ratios, Cosine, Sine, Tangent

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What are Trigonometric Equations?

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Trigonometric Equation is a form of algebraic equation that involves one or more trigonometric ratios of unknown angles. There are three primary trigonometric ratios, namely sin, cos, and tan. 

• The three other trigonometric ratios, sec, cosec, and cot, are the reciprocals of sin, cos, and tan, respectively.
• All trigonometric equations are solved for the value of θ.
• A solution can be easily calculated for an equation by drawing a graph.
• The linear equation of the form ax + b = 0 can be represented in the trigonometry equation as aSinθ + b = 0.
• The identities of a trigonometric function do not refer to a particular figure.

Examples of trigonometric equations are:

• Sin4x - Sin6x + Sin2x = 0
• 2Cos2x + 4Sinx = 0
• Cos4x = Cos2x

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Trigonometric Equations and Their Solutions

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Equations containing trigonometric functions of a single variable are known as Trigonometric Equations. For Example: cos²x + 5 cosx – 7 = 0 .

• The solutions of equations for a trigonometric function in variable x, where x lies between 0 ≤ x ≤ 2π, are the principal solutions. 
• If the solution has the integer ‘n’, it is known as the general solution.
• Trigonometric equations are solved for right-angled triangles. 
• The domain of the trigonometric function must be assumed first before determining a valid solution.
• To solve an equation, check for patterns, factors and common denominators.

The table below explains the solutions for numerous trigonometric equations.

Let us consider the proof for the equation sin x = sin y implies x = nπ + (-1)ⁿ y, where n € Z and x and y are any real numbers.

• Suppose sin x = sin y, then sin x – sin y =0 i.e. 2 cos(x+y)/2 sin(x-y)/2 =0
• It outputs value is equivalent to cos (x+y)/2 = 0 or (x+y)/2 = (2n + 1) π/2
• In similar way, sin (x-y)/2 = 0 or (x-y)/2 = nπ which implicates x = (2n+1) π – y; or x= 2nπ +y
• Thus, x = (2n+1) π + (-1)²ⁿ⁺¹ y; or x = 2nπ + (-1)²ⁿ y
• Adding these two outputs, we obtain x = nπ + (-1)ⁿ y, where n ∈ Z.

Hence proving the equation sin x = sin y

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Proofs of Trigonometric Equations

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Here are the proofs of solutions of all trigonometric equations: 

If x and y are real numbers, sin x = sin y implies x = nπ + (-1)ⁿ y, where n is an integer.

Suppose sin x = sin y, then sin x - sin y = 0.

• Using the formula, sin A - sin B = 2 cos (A+B)/2 .sin (A-B)/2
• This equation can be transformed into 2 cos (x+y)/2. sin (x-y)/2 = 0.
• It can further be simplified as either cos (x+y)/2 = 0 or sin (x-y)/2 = 0.
• Knowing that cos A = 0 when A = (2n+1)π/2 and sin A = 0 when A = nπ, where n is an integer, this means that (x+y)/2 = (2n+1)π/2 or (x-y)/2 = nπ.
• Substituting the values of (x+y)/2 and (x-y)/2 in the above step, x = (2n+1)π - y or x = 2nπ + y.

Combining these results, x = nπ + (-1)ⁿ y, where n is an integer.

If x and y are real numbers, cos x = cos y implies x = 2nπ ± y, where n is an integer:

Suppose cos x = cos y, then cos x - cos y = 0.

• Using the formula, cos A - cos B = -2 sin (A+B)/2 * sin (A-B)/2, this can be transformed into -2 sin (x+y)/2 .sin (x-y)/2 = 0.
• This can further be simplified as either sin (x+y)/2 = 0 or sin (x-y)/2 = 0.
• Knowing that sin A = 0 when A = nπ, where n is an integer, this means that (x+y)/2 = nπ or (x-y)/2 = nπ.
• Substituting the values of (x+y)/2 and (x-y)/2 in the above step, x = 2nπ - y or x = 2nπ + y.

Hence, x = 2nπ ± y, where n is an integer.

If x and y are not odd multiples of π/2, then tan x = tan y implies x = nπ + y, where n is an integer.

Suppose tan x = tan y, then tan x - tan y = 0.

• Using the formula, tan x - tan y = (sin x / cos x - sin y / cos y) / (cos x cos y), this can be transformed into (sin x cos y - cos x sin y) / (cos x cos y) = 0.
• Using the formula, sin (A - B) = sin A cos B - sin B cos A, this can be further simplified as sin (x-y) / (cos x cos y) = 0.
• Knowing that sin A = 0 when A = nπ, where n is an integer, this means that sin (x-y) = 0.

Hence, x - y = nπ, where n is an integer and finally, x = nπ + y, where n is an integer.

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Solving Trigonometric Equations

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Trigonometric equations differ from normal algebraic equations. In trignometric equation, the number of solutions is based on the angle of the trigonometric function rather than the degree of the variable.

• For instance, the equation 2cosθ - 1 = 0 has a solution of cosθ = 1/2, which corresponds to angles of π/3, 5π/3, 7π/3, 11π/3, and so forth.
• These values repeat after every 2π radians and cos x is positive in the first and fourth quadrants.

To solve trigonometric equations, two types of solutions exist: 

Principal Solution

Principal Solution refers to the initial values of angles for the trigonometric functions. 

• The solutions of the sinθ and cosθ equations repeat after an interval of 2π, while the solution of the tanθ equation repeats after an interval of π. 
• The solutions for x values between 0 and 2π are referred to as the principal solutions.
• For example principal solution of sinθ = ½ are π/6 and 5π/6 as these two solutions lie between 0 to 2π.

General Solution

General Solution refers to the consolidated values of the angles for the same answer of the trigonometric function. The solutions beyond 2π are expressed as a general solution. The general solutions for Sinθ, Cosθ, and Tanθ are as follows:

• For example general solution of sinθ = ½ is 5π/6 as this solution lie beyond 2π.
• Sinθ = Sinα, with the general solution being θ = nπ + (-1)nα, where n ∈ Z
• Cosθ = Cosα, with the general solution being θ = 2nπ + α, where n ∈ Z
• Tanθ = Tanα, with the general solution being θ = nπ + α, where n ∈ Z.

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Steps to Solve Trigonometric Equations

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The process of solving a trigonometric equation requires the following steps:

Simplification

The process followed in case of simplification is as follows:

• Simplify the equation by transforming it into an equation with only one trigonometric function, such as sin, cos, or tan.
• This may involve converting multiple angles or submultiple angles into a single angle.

Polynomial or Equation Form

In this step write the equation in polynomial or equation form, for instance, as a linear, quadratic, or polynomial equation.

Solving the Equation

​The process followed in case of solving an equation is as follows:

• Solve the equation as you would solve any other algebraic equation.
• Use the methods of algebra, such as factoring, completing the square, or using the quadratic formula.

Finding the Angle

​The process followed in case of finding the angle is as follows:

• Once the equation is solved, the value of the trigonometric function represents the solution of the equation.
• The angle of the trigonometric function can be found using the inverse functions, such as inverse sine, inverse cosine, or inverse tangent.

Principal Solution

​The process followed in case of principle solution is as follows:

• Remember to consider the restriction of the principal solution while finding the solution of the equation.
• The initial values of angles for trigonometric functions are referred to as the principal solutions.
• These solutions repeat after an interval of 2π for sine and cosine and π for tangent.

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Examples of Solving Trigonometric Equations

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Here are some solved examples on Trigonometric equations: 

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Trigonometric Ratios

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Trigonometry is built on three main functions which are sine (sin), cosine (cos), and tangent (tan). These functions take an angle as input and return a value that represents a ratio of the sides of a right triangle. 

• The three formulas of trignnometry are collectively known by the abbreviation SohCahToa
• It is based on the concept of Pythagoras Theorem.

Sine Function

The sine function of an angle is the ratio of the side opposite the angle to the hypotenuse (the longest side) of a right triangle.

Cosine Function

The cosine function of an angle is the ratio of the adjacent side (the side next to the angle) to the hypotenuse of a right triangle.

Tangent Function

The tangent function of an angle is the ratio of the opposite side to the adjacent side of a right triangle.

Assume a right-angled triangle (as shown below in figure). From the figure, ∠A made is an acute angle (less than 90).

Right-Angled Triangle

From the figure, the following trigonometric ratios are formed

Trigonometric Ratios

Each of these trigonometric ratios can be calculated at various angles such as 30, 60, 90, and 180. few of the important standard angles at values are mentioned below in the table.

Trigonometric Ratios of Standard Angles can be depicted in a tabular form as shown below:

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Pythagoras Theorem

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Pythagoras Theorem states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. This theorem is used to find the length of a missing side of a right triangle.

• We know that the right triangle ABC applies Pythagoras property as AB² = AC² + BC². By Dividing into both sides with AB², we get, 1 = cos²A + sin²A.
• Similarly, we can obtain other identities which are as follows:

sin²A + cos²A = 1

sec²A – tan²A = 1

cosec²A – cot²A = 1

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Trigonometric Ratios of Complementary Angles

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Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. One of the key concepts in trigonometry is the idea of complementary angles, which are pairs of angles that add up to 90 degrees. 

• In trigonometry, it is often useful to know the trigonometric ratios of complementary angles, as these ratios can be used to solve problems involving triangles. 
• Trigonometric ratios of complementary angles are simply the trigonometric ratios of the two angles in a complementary pair. 
• For example, if one angle in a complementary pair is 30 degrees, then the other angle must be 60 degrees.
• The trigonometric ratios for these two angles can be found and used in calculations. 
• These ratios are important in a variety of fields, including engineering, physics, and navigation.
• They provide a foundation for solving more complex problems.

We know that ∠A + ∠B = 90° or ∠B = 90° – ∠A. let us consider the value for the sine of ∠B, be sin B = sin ( 90° – ∠A ) = AC/AB. But AC/AB is equal to cos A.

Thus, we can obtain sin (90° – ∠A) = cos A. Similarly, we can calculate other trigonometric ratios for complementary angles as follows:

• sin (90°– ∠A) = cos A
• cos (90°– ∠A) = sin A
• tan (90°– ∠A) = cot A
• cot (90°– ∠A) = tan A
• sec (90°– ∠A) = cosec A
• cosec (90°– ∠A) = sec A

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Trigonometry Identities

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Trigonometric Identities are mathematical equations that relate the trigonometric functions to one another. Some of the most commonly used identities include 

• sin²θ + cos²θ = 1 
• tan θ= sinθ/cosθ

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Trigonometric Formulas

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Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. One of the key tools in trigonometry is a set of formulas that describe the relationships between the sides and angles of triangles. 

• These formulas, known as trigonometric formulas, are used to solve a wide range of problems in mathematics, science, and engineering. 
• Some of the most commonly used trigonometric formulas include the Pythagorean theorem, the formulas for the sine, cosine, and tangent functions. 

Trigonometric functions (sine, cosine, tangent, etc.) are important mathematical tools for solving problems in fields such as mathematics, physics, engineering, and more. 

Here is a list of some of the basic trigonometric equations and formulas:

Cosine and Sine of a Sum or Difference of Angles

Some of the equation and formulas for sum or difference of angles are as follows:

• cos (x + y) = cos x cos y – sin x sin y
• sin (x + y) = sin x cos y + cos x sin y
• cos (x - y) = cos x cos y + sin x sin y
• sin (x - y) = sin x cos y - cos x sin y

Double Angle Formulas

Some of the equation and formulas for double angles are as follows:

• cos ²x = cos² x - sin² x = 2 cos² x - 1 = 1 - 2 sin² x = 1 - tan² x / (1 + tan² x)
• sin ²x = 2 sin x cos x = 2 tan x / (1 + tan² x)

Half-Angle Formulas

Some of the equation and formulas for half angles are as follows:

• sin (x/2) = ±√[(1 - cos x) / 2]
• cos (x/2) = ±√[(1 + cos x) / 2]

Sum-to-Product Formulas

Some of the equation and formulas for sum to product are as follows:

• cos x + cos y = 2 cos ( (x + y)/2 ) cos ( (x - y)/2 )
• cos x - cos y = -2 sin ( (x + y)/2 ) sin ( (x - y)/2 )
• sin x + sin y = 2 sin ( (x + y)/2 ) cos ( (x - y)/2 )
• sin x - sin y = 2 cos ( (x + y)/2 ) sin ( (x - y)/2 )

Trigonometric Functions of Complementary Angles

Some of the equation and formulas for complementary angles are as follows:

• cos (π - x) = -cos x
• sin (π - x) = sin x
• cos (π + x) = -cos x
• sin (π + x) = -sin x

Trigonometric Functions of Multiples of π

Some of the equation and formulas for a multiple of π are as follows:

• cos (2nπ + x) = cos x
• sin (2nπ + x) = sin x
• cos (-x) = cos x
• sin (-x) = -sin x

Trigonometric Functions of Angles with a Measure of π/2

Some of the equation and formulas for a multiple of π/2 are as follows:

• cos (π/2 + x) = -sin x
• sin (π/2 + x) = cos x

Conversion between Degree and Radian Measures

The formula for conversion between degree and radian are as follows:

• Radian Measure = Degree Measure x π/180
• Degree Measure = Radian Measure x 180/π

Periodicity Identities (in Radians)

Some of the formulas for a trigonometry in terms of radians are as follows:

• sin (π/2 – A) = cos A & cos (π/2 – A) = sin A
• sin (π/2 + A) = cos A & cos (π/2 + A) = – sin A
• sin (3π/2 – A) = – cos A & cos (3π/2 – A) = – sin A
• sin (3π/2 + A) = – cos A & cos (3π/2 + A) = sin A
• sin (π – A) = sin A & cos (π – A) = – cos A
• sin (π + A) = – sin A & cos (π + A) = – cos A
• sin (2π – A) = – sin A & cos (2π – A) = cos A
• sin (2π + A) = sin A & cos (2π + A) = cos A

Cofunction Identities (in Degrees)

Some of the formulas for a trigonometry in terms of degrees are as follows:

• sin(90°−x) = cos x
• cos(90°−x) = sin x
• tan(90°−x) = cot x
• cot(90°−x) = tan x
• sec(90°−x) = cosec x
• cosec(90°−x) = sec x

Sum & Difference Identities

Some of the formulas for a trigonometry for sum and difference are as follows:

• sin(x+y) = sin(x)cos(y)+cos(x)sin(y)
• cos(x+y) = cos(x)cos(y)–sin(x)sin(y)
• tan(x+y) = tan x+tan y/1-tan x.tan y
• sin(x–y) = sin(x)cos(y)–cos(x)sin(y)
• cos(x–y) = cos(x)cos(y) + sin(x)sin(y)
• tan(x-y) = tan x-tan y/1+tan x.tan y

Inverse Trigonometry Formulas

Some of the formulas for a inverse trigonometry are as follows:

• sin^-1 (–x) = – sin^-1 x
• cos^-1 (–x) = π – cos^-1 x
• tan^-1 (–x) = – tan^-1 x
• cosec^-1 (–x) = – cosec^-1 x
• sec^-1 (–x) = π – sec^-1 x
• cot^-1 (–x) = π – cot^-1 x

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Things to Remember

• Trigonometric Equation is an equation involving one or more trigonometric ratios of unknown angles.
• Trigonometry is based on three main functions namely sine (sin), cosine (cos), and tangent (tan), which represent ratios of sides of a right triangle.
• Trigonometric Ratios can be calculated for standard angles such as 30, 60, 90, and 180 degrees.
• If two real numbers x and y satisfy the equation sin x = sin y, then it follows that x = nπ + (-1)ⁿ y, where n is an integer.
• In the case where cos x = cos y, the conclusion is that x = 2nπ ± y, again with n being an integer. 
• If x and y are not odd multiples of π/2, and if tan x = tan y, then x = nπ + y, where n is an integer.
• It's important to note that when sin A equals 0, A must equal nπ, with n being an integer. 
• If cos A equals 0, A must equal (2n + 1)π/2, with n being an integer.

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