Trigonometric Equations 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.

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.