Identities of Trigonometry and Trigonometric Equations

In trigonometry, an identity is an equality involving a trigonometric function that is valid for every possible value of the variables involved and ensures that both sides of the equality are defined. Trigonometric identities can be defined as follows: Trigonometric identities are going to be the topic of discussion during this brief lecture. Sin, cos, and tan are the three fundamental ratios that are used in trigonometry. The reciprocals of sin, cos, and tan are represented by the trigonometric ratios sec, cosec, and cot, respectively. These are the three additional ratios that are used in trigonometry. What are the relationships between these trigonometric ratios (sin, cos, tan, sec, cosec, and cot)? Trigonometric identities are the connecting factor between them (or in short trig identities). In the following parts, let’s go further into the trigonometric identities and make sure we fully grasp them.

What are Trigonometric Identities

Trigonometric identities are equations that relate to various trigonometric functions and hold true for any value of the variable that may be present in the domain. These equations are known as “trigonometric identities.” An identity, in its most fundamental form, is an equation that is valid for any and all possible values of the variable (or variables) that it contains.

For instance, some of the identities in algebra are as follows:

(a + b)² = a² + 2ab + b²

(a – b)² = a² – 2ab+ b²

(a + b)(a-b)= a² – b²

The algebraic identities only relate the variables, but the trigonometric identities only relate the six trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent.

Trigonometric Equations

The term “trigonometry” comes from two Greek words: trigonon, which means “triangle,” and metron, which means “measure” (measure). Triangle geometry is a subfield of mathematics that investigates the connections that exist between the lengths of a triangle’s sides and the angles formed by those sides. An equation using trigonometric ratios of unknown angles is known as a trigonometric equation. Trigonometric equations might involve one or more unknown angle ratios. It is stated as ratios of sine angles (sin), cosine angles (cos), tangent angles (tan), cotangent angles (cot), secant angles (sec), and cosecant angles (cosec). For instance, an example of a trigonometric equation might be cos2 x + 5 sin x = 0. The term “solutions” refers to any and all feasible value combinations that are capable of making the specified trigonometric equation true.

Reciprocal Identities

The following is a list of the reciprocal identities:

cosec θ = 1/sin θ

sec θ = 1/cos θ

cot θ = 1/tan θ

sin θ = 1/cosec θ

cos θ = 1/sec θ

tan θ = 1/cot θ

All of them are derived from a triangle with an acute angle in one of its sides. Using trigonometric formulas, it is possible to determine the values of the sine, cosine, tangent, secant, cosecant, and cotangent of a right triangle once the height and base side of the triangle has been determined. Utilizing the various trigonometric functions allows for the derivation of the reciprocal trigonometric identities as well.

Pythagorean Trigonometric Identities

The Pythagorean theorem is the starting point for deriving the trigonometric identities known as the Pythagorean identities. The following is what we obtain when we apply the Pythagorean theorem to the triangle with a right angle:

Opposite² + Adjacent² = Hypotenuse²

Using the Hypotenuse² as a divider between the two sides

Hypotenuse²/Hypotenuse² equals the combination of the opposite²/Hypotenuse² and adjacent²/Hypotenuse².

sin²θ + cos²θ = 1

This is one of the identities attributed to Pythagoras. We can obtain two more Pythagorean trigonometric identities by proceeding in the same manner as before.

1 + tan²θ = sec²θ

1 + cot²θ = cosec²θ

Periodicity Identities (in Radians)

These formulas are utilised in order to rotate the angles by a value of  π/2, π, 2π,  etc. These identities can also be referred to as co-function identities.

• 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

All trigonometric identities are cyclic in nature. They do the same thing over and over again following this periodicity constant. This periodicity constant varies depending on which particular trigonometric identity is being used. However, this is not the case for cos 45° and cos 225°, even if tan 45° = tan 225°. To verify the values, please refer to the trigonometry table that was just provided.

Co-function Identities (in Degrees)

It is also possible to represent the co-function or periodic identities using degrees, 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

• 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)= tanx+tany/(1-tanxtany)

• 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)= tanx-tany/(1+tanxtany)

Double Angle Identities

• sin 2A = 2 sin A cos A = 2 tan A/(1+tan²A)

• cos 2A = cos²A – sin²A  = 2cos²A-1 = 1-2 sin²A = (1-tan²A)/(1+tan²A)

• tan 2A = 2 tan A/(1-tan²A)

• sec2A= 2sec²A/(2-sec²A)

• cosec2A= secA.cosecA/2

Triple Angle Identities

• tan 3A = (3 tan A- tan³A)/(1-3tan²A)

• Sin 3A = 3sin A – 4sin³A

• Cos 3A = 4cos³A-3cos A

Identities for the Sum, Difference, and Product of Trigonometric Ratios

• cos (A + B) = cos A cos B – sin A sin B

• cos (A – B) = cos A cos B + sin A sin B

• tan (A + B) = (tan A + tan B)/ (1 – tan A tan B)

• tan (A – B) = (tan A – tan B)/ (1 + tan A tan B)

• cot (A + B) = (cot A cot B – 1)/(cot B – cot A)

• cot (A – B) = (cot A cot B + 1)/(cot B – cot A)

• 2 sin A⋅cos B = sin(A + B) + sin(A – B)

• 2 cos A⋅cos B = cos(A + B) + cos(A – B)

• 2 sin A⋅sin B = cos(A – B) – cos(A + B)

Conclusion

Trigonometry equations that are always true are called trig identities, and they are frequently employed in the process of solving problems in trigonometry and geometry as well as comprehending a variety of mathematical aspects. The ability to recall and comprehend significant mathematical principles, as well as the ability to solve a wide variety of mathematical problems, all depend on your familiarity with key trig identities.

A trigonometric equation is one that involves one or more unknown trigonometric ratios. It is stated as ratios of sine angles (sin), cosine angles (cos), tangent angles (tan), cotangent angles (cot), secant angles (sec), and cosecant angles (cosec).