Introduction- Trigonometric Equations

An equation involving one or more trigonometric ratios of unknown angles is called a trigonometric equation. A trigonometric equation can be written as two expressions equating to form a relationship. Both the expressions include trigonometric functions such as sinx, cosx, tax, and so on.

Example: The equation cos2x – 4 sin x = 1 is a trigonometric equation. Because it is not fulfilled across all values of x, it is a trigonometric equation rather than an identity., e.g., the equation is not satisfied at (2n + 1)π/4. 

Simple Trigonometric Equations and Their Solutions:

The term “solutions of the given equation” refer to all potential values of the variable that fulfill the given equation. All feasible values fulfilling the equation must be acquired for a full solution. When attempting to solve trigonometric equations, we look for all sets of values that fulfill the given equation. Occasionally, with simple equations or when it is possible to build a graph of an equation, the answer can be found simply by looking at the graph.

The general solution of the equation sin θ = k. 

It is known that if sin θ = k, k must lie between –1 ≤ k ≤ 1 

To compute α ∈ [–π/2, π/2]

Now, sin (-π)/2 = -1 & sin π/2 = 1, for sin θ = k, such that α = sin-1k

sin θ = sin α, α ∈ [–π/2, π/2]

⇒ sin θ – sin α = 0

⇒ 2 sin {(θ – α)/2} cos {θ + α)/2} = 0 

For the given equation to be true, either sin {(θ – α)/2) = 0

or ((θ – α)/2) = integral multiple of π

∴ θ – α = 2nπ

i.e. θ = 2nπ + α

θ = 2nπ + (–1)2n α where n = 0, ±1, ±2 … (1) 

or, cos {(θ + α)/2} = 0

i.e. {(θ + α)/2} = any odd multiple of π/2

i.e. {(θ + α)/2} = (2n + 1)π/2 

i.e. θ = (2n + 1)π – α 

⇒ θ = (2n +1)π + (–1)2n+1 α … (2)

Concluding from (1) and (2)

θ = nπ + (–1)n α, where n is integral multiple, is the general solution of the equation sin θ = k 

Trigonometric Equations with their general Solutions:

Compound Angle Formula

A mathematical sum of two or more angles yields a compound angle. Using trigonometric functions, we may denote compound angles using trigonometric identities. In trigonometry, one can use the compound angle formula or the addition formula to compute the sum and difference of functions. Here, we’ll look at functions like (x+y) and (x-y). The following is the formula for trigonometric ratios of compound angles:

sin (x + y) = sin x cos y + cos x sin y

sin (x – y) = sin x cos y – cos x sin y

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

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

tan (x + y) = [tan x + tan y] / [1 – tan x tan y]

tan (x – y) = [tan x – tan y] / [1 + tan x tan y]

sin(x + y) sin(x – y) = sin 2x – sin 2y = cos 2y – cos 2x

cos(x + y) cos(x – y) = cos 2x – sin 2x – sin 2y = cos 2y – sin 2x

Steps to Solve Trigonometric Equations:

Following tips and steps will help you systematically solve trigonometric equations.

1. Try to simplify the problem into a single trigonometric ratio, ideally sin or cos. When converting a problem to a sine or cosine form, a cosine form is more efficient than a sine form. This is because the general solution of sine might include (–1)n, which is difficult when contrasted positive form generated in cosine form.

2. Next, factor the polynomial using these ratios.

3. Solve for each component until LHS equals zero. And, depending on the previously predicted outcomes, put down general answers for each element.

e.g. sin θ – k1 = 0 ⇒ θ = nπ + (–1)n sin-1 k1

cos θ – k2 = 0 ⇒ θ = 2nπ + cos–1 k2. 

Note: One should always verify the solution by confirming that the value of k lies between -1 and +1.

Conclusion

Trigonometric equations are mathematical equations that include trigonometric functions. The term “solutions of the given equation” refer to all potential values of the unknown that fulfill the given equation. When solving a trigonometric equation, we seek all sets of values that fulfill the given equation. A compound angle is the mathematical sum of two or more angles. We may indicate compound angles using trigonometric identities by employing trigonometric functions. To compute the sum and difference of functions in trigonometry, use the compound angle formula or the addition formula.