Trigonometric Identities

The relationship between trigonometric functions is expressed via trigonometric identities, sometimes referred to as trig identities or trig formulae. They remain true for all real number values of the assigned variables in them.

• Reciprocal Identities
• Pythagorean Identities
• Periodicity Identities (in Radians)
• Even and Odd Angle Formula
• Cofunction identities (in Degrees)
• Sum and Difference Identities
• Double Angle Identities
• Inverse Trigonometry Formulas
• Triple Angle Identities
• Half Angle Identities
• Sum to Product Identities
• Product Identities

Let’s discuss these identites in detail.

Reciprocal Identities

All of the reciprocal identities are obtained using a right-angled triangle as a reference. Reciprocal Identities are as follows:

• 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

Pythagorean Identities

According to the Pythagoras theorem, in a right triangle, if ‘c’ is the hypotenuse and ‘a’ and ‘b’ are the two legs then c2 = a2 + b2. We can obtain Pythagorean identities using this theorem and trigonometric ratios. We use these identities to convert one trig ratio into other.

• sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ
• 1 + cot²θ = cosec²θ

Periodicity Identities (in Radians)

These identities can be used to shift the angles by π/2, π, 2π, etc. These are also known as co-function identities.

All trigonometric identities repeat themselves after a particular period. Hence are cyclic in nature. This period for the repetition of values is different for different trigonometric 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

Here’s a table that compares the trigonometric properties in different quadrants :

Even and Odd Angle Formula

The Even and Odd Angle Formulas , also known as Even-Odd Identities are used to express trigonometric functions of negative angles in terms of positive angles. These trigonometric formulas are based on the properties of even and odd functions.

• sin(-θ) = -sinθ
• cos(-θ) = cosθ
• tan(-θ) = -tanθ
• cot(-θ) = -cotθ
• sec(-θ) = secθ
• cosec(-θ) = -cosecθ

Cofunction identities (in Degrees)

Cofunction identities give us the interrelationship between various trigonometry functions. The co-function are listed here in degrees:

• 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 and Difference Identities

The sum and difference identities are the formulas that relate the sine, cosine, and tangent of the sum or difference of two angles to the sines, cosines, and tangents of the individual 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)sin(y)
• cos(x-y)=cos(x)cos(y) + sin(x)sin(y
• \[Tex]\bold{\tan(x+y)=\frac{tan\text{ x}+tan\text{ y}}{1- tan\text{ x}.tan\text{ y}}}[/Tex]
• [Tex]\bold{\\tan(x -y)=\frac{tan\text{ x}-tan\text{ y}}{1+ tan\text{ x}.tan\text{ y}}}[/Tex]

Double Angle Identities

Double angle identities are the formulas that express trigonometric functions of angles which are double the measure of a given angle in terms of the trigonometric functions of the original angle.

• sin (2x) = 2sin(x) • cos(x) = [2tan x/(1 + tan² x)]
• cos (2x) = cos²(x) – sin²(x) = [(1 – tan² x)/(1 + tan² x)] = 2cos²(x) – 1 = 1 – 2sin²(x)
• tan (2x) = [2tan(x)]/ [1 – tan²(x)]
• sec (2x) = sec² x/(2 – sec² x)
• cosec (2x) = (sec x • cosec x)/2

Inverse Trigonometry Formulas

Inverse trigonometry formulas relate to the inverse trigonometric functions, which are the inverses of the basic trigonometric functions. These formulas are used to find the angle that corresponds to a given trigonometric ratio.

• 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

Triple Angle Identities

Triple Angle Identities are formulas used to express trigonometric functions of triple angles (3θ) in terms of the functions of single angles (θ). These trigonometric formulas are useful for simplifying and solving trigonometric equations where triple angles are involved.

sin 3x=3sin x – 4sin³x

cos 3x=4cos³x – 3cos x

[Tex]\bold{\tan \text{ 3x}=\frac{3 tan\text{ x}-tan^3x}{1- 3tan^2x}}[/Tex]

Half Angle Identities

Half-angle identities are those trigonometric formulas that are used to find the sine, cosine, or tangent of half of a given angle. These formulas are used to express trigonometric functions of half-angles in terms of the original angle.

[Tex]\bold{\sin\frac{x}{2}=\pm \sqrt{\frac{1- cos\text{ x}}{2}}}[/Tex]

[Tex]\bold{cos\frac{x}{2}=\pm \sqrt{\frac{1+ cos\text{ x}}{2}}}[/Tex]

[Tex]\bold{\tan(\frac{x}{2})=\pm \sqrt{\frac{1- cos(x)}{1+cos(x)}}}[/Tex]

Also,

[Tex]\bold{\tan(\frac{x}{2})=\pm \sqrt{\frac{1- cos(x)}{1+cos(x)}}}[/Tex]

[Tex]\bold{\tan(\frac{x}{2})=\pm \sqrt{\frac{(1- cos(x))(1-cos(x))}{(1+cos(x))(1-cos(x))}} }[/Tex]

[Tex]\bold{=\sqrt{\frac{(1- cos(x))^2}{1-cos^2(x)}} }[/Tex]

[Tex]\bold{=\sqrt{\frac{(1- cos(x))^2}{sin^2(x)}} }[/Tex]

[Tex]\bold{=\frac{1-cos(x)}{sin(x)}}[/Tex]

[Tex]\bold{\tan(\frac{x}{2})=\frac{1-cos(x)}{sin(x)}}[/Tex]

Sum to Product Identities

Sum to Product identities are the trigonometric formulas that help us to express sums or differences of trigonometric functions as products of trigonometric functions.

• sinx + siny = 2[sin((x + y)/2)cos((x − y)/2)]
• sinx − siny = 2[cos((x + y)/2)sin((x − y)/2)]
• cosx + cosy = 2[cos((x + y)/2)cos((x − y)/2)]
• cosx − cosy = −2[sin((x + y)/2)sin((x − y)/2)]

Product Identities

Product identities, also known as product-to-sum identities are the formulas that allow the expression of products of trigonometric functions as sums or differences of trigonometric functions.

These trigonometric formulas are derived from the sum and difference formulas for sine and cosine.

• sinx⋅cosy = [sin(x + y) + sin(x − y)]/2
• cosx⋅cosy = [cos(x + y) + cos(x − y)]/2
• sinx⋅siny = [cos(x − y) − cos(x + y)]/2

Trigonometry formulas are equations that relate the sides and angles of triangles. They are essential for solving a wide range of problems in mathematics, physics, engineering and other fields.

Here are some of the most common types of trigonometry formulas:

• Basic definitions: These formulas define the trigonometric ratios (sine, cosine, tangent, etc.) in terms of the sides of a right triangle.
• Pythagorean theorem: This theorem relates the lengths of the sides in a right triangle.
• Angle relationships: These formulas relate the trigonometric ratios of different angles, such as sum and difference formulas, double angle formulas, and half angle formulas.
• Reciprocal identities: These formulas express one trigonometric ratio in terms of another, such as sin(θ) = 1/coc(θ).
• Unit circle: The unit circle is a graphical representation of the trigonometric ratios, and it can be used to derive many other formulas.
• Law of sines and law of cosines: These laws relate the sides and angles of any triangle, not just right triangles.

Read on to learn about different trigonometric formulas and identities, solved examples, and practice problems.