Trigonometry Formulas – List of All Trigonometric Identities and Formulas

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.

What is Trigonometry?

Trigonometry is defined as a branch of mathematics that focuses on the study of relationships involving lengths and angles of triangles. Trigonometry consists of different kinds of problems which can be solved using trigonometric formulas and identities.

Angles (In Degrees)	0°	30°	45°	60°	90°	180°	270°	360°
Angles (In Radians)	0°	π/6	π/4	π/3	π/2	π	3π/2	2π
sin	0	1/2	1/√2	√3/2	1	0	-1	0
cos	1	√3/2	1/√2	1/2	0	-1	0	1
tan	0	1/√3	1	√3	∞	0	∞	0
cot	∞	√3	1	1/√3	0	∞	0	∞
cosec	∞	2	√2	2/√3	1	∞	-1	∞
sec	1	2/√3	√2	2	∞	-1	∞	1
Trigonometry Ratios Table

Trigonometry Functions

Trigonometric functions are mathematical functions that relate angles of a right triangle to the lengths of its sides. They have wide applications across various fields such as physics, engineering, astronomy, and more. The primary trigonometric functions include sine, cosine, tangent, cotangent, secant, and cosecant.

Trigonometric Function	Domain	Range	Period
sin(θ)	All Real Number i.e., R	[-1, 1]	2π or 360°
cos(θ)	All Real Numbers i.e.,	[-1, 1]	2π or 360°
tan(θ)	All Real Numbers excluding odd multiples of π/2	R	π or 180°
cot(θ)	All Real Numbers excluding multiples of π	R	2π or 360°
sec(θ)	All Real Numbers excluding values where cos(x) = 0	R-[-1, 1]	2π or 360°
cosec(θ)	All Real Numbers excluding multiples of π	R-[-1, 1]	π or 180°

Trigonometry Formula Overview

Trigonometry formulas are mathematical expressions that relate the angles and sides of a Right Triangle. There are 3 sides a right-angled triangle is made up of:

• Hypotenuse: This is the longest side of a right-angled triangle.
• Perpendicular/Opposite side: It is the side that forms a right angle with respect to the given angle.
• Base: The base refers to the adjacent side where both the hypotenuse and the opposite side are connected.

Trigonometry Ratio

All the trigonometric ratios, product identities, half angle formulas, double angle formulas, sum and difference identities, cofunction identities, a sign of ratios in different quadrants, etc. are briefly given here for the students of Classes 9, 10, 11, 12.

Here is the list of formulas in trigonometry we are going to discuss:

• Basic Trigonometric Ratio Formulas
• Unit Circle Formulas
• Trigonometric Identities

Basic Trigonometric Ratios

There are 6 ratios in trigonometry. These are referred to as Trigonometric Functions. Below is the list of trigonometric ratios, including sine, cosine, secant, cosecant, tangent and cotangent.

List of Trigonometric Ratios
Trigonometric Ratio	Definition
sin θ	Perpendicular / Hypotenuse
cos θ	Base / Hypotenuse
tan θ	Perpendicular / Base
sec θ	Hypotenuse / Base
cosec θ	Hypotenuse / Perpendicular
cot θ	Base / Perpendicular

Unit Circle Formula in Trigonometry

For a unit circle, for which the radius is equal to 1, θ is the angle. The values of the hypotenuse and base are equal to the radius of the unit circle.

Hypotenuse = Adjacent Side (Base) = 1

The ratios of trigonometry are given by:

• sin θ = y/1 = y
• cos θ = x/1 = x
• tan θ = y/x
• cot θ = x/y
• sec θ = 1/x
• cosec θ = 1/y

Trigonometric Functions Diagram

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

Trigonometry Formulas Chart

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 :

Quadrant	Sine (sin θ)	Cosine (cos θ)	Tangent (tan θ)	Cosecant (csc θ)	Secant (sec θ)	Cotangent (cot θ)
I (0° to 90°)	Positive	Positive	Positive	Positive	Positive	Positive
II (90° to 180°)	Positive	Negative	Negative	Positive	Negative	Negative
III (180° to 270°)	Negative	Negative	Positive	Negative	Negative	Positive
IV (270° to 360°)	Negative	Positive	Negative	Negative	Positive	Negative

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]\tan(x+y)=\frac{tan\text{ x}+tan\text{ y}}{1- tan\text{ x}.tan\text{ y}}[/Tex]
• [Tex]\\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]\\\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]\\\sin\frac{x}{2}=\pm \sqrt{\frac{1- cos\text{ x}}{2}}[/Tex]

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

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

Also,

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

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

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

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

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

[Tex]\\\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

List of Trigonometry Formulas

The table given below consists of basic trigonometry ratios for angles such as such as 0°, 30°, 45°, 60°, and 90° that are commonly used for solving problems.

Table of Trigonometric Ratios
Angles(In Degrees)	0	30	45	60	90	180	270	360
Angles(In Radians)	0	π/6	π/4	π/3	π/2	π	3π/2	2π
sin	0	1/2	1/√2	√3/2	1	0	-1	0
cos	1	√3/2	1/√2	1/2	0	-1	0	1
tan	0	1/√3	1	√3	∞	0	∞	0
cot	∞	√3	1	1/√3	0	∞	0	∞
cosec	∞	2	√2	2/√3	1	∞	-1	∞
sec	1	2/√3	√2	2	∞	-1	∞	1

Solved Questions on Trigonometry Formula

Here are some solved examples on trigonometry formulas to help you get a better grasp of the concepts.

Question 1: If cosec θ + cot θ = x, find the value of cosec θ – cot θ, using trigonometry formula.

Solution:

cosec θ + cot θ = x

We know that cosec²θ+ cot²θ = 1

(cosec θ -cot θ)( cosec θ+ cot θ) = 1

(cosec θ -cot θ) x = 1

cosec θ -cot θ = 1/x

Question 2: Using trigonometry formulas, show that tan 10° tan 15° tan 75° tan 80° =1

Solution:

We have, 

L.H.S= tan 10° tan 15° tan 75° tan 80°

= tan(90-80)° tan 15° tan(90-15)° tan 80°

= cot 80° tan 15° cot 15° tan 80°

=(cot 80° * tan 80°)( cot 15° * tan 15°)

= 1 = R.H.S

Question 3: If sin θ cos θ = 8, find the value of (sin θ + cos θ)² using the trigonometry formulas.

Solution:

(sin θ + cos θ)²

= sin²θ + cos²θ + 2sinθcosθ

= (1) + 2(8) = 1 + 16 = 17

= (sin θ + cos θ)² = 17

Question 4: With the help of trigonometric formulas, prove that (tan θ + sec θ – 1)/(tan θ – sec θ + 1) = (1 + sin θ)/cos θ.

Solution:

L.H.S = (tan θ + sec θ – 1)/(tan θ – sec θ + 1)

= [(tan θ + sec θ) – (sec²θ – tan²θ)]/(tan θ – sec θ + 1), [Since, sec²θ – tan²θ = 1]

= {(tan θ + sec θ) – (sec θ + tan θ) (sec θ – tan θ)}/(tan θ – sec θ + 1)

= {(tan θ + sec θ) (1 – sec θ + tan θ)}/(tan θ – sec θ + 1)

= {(tan θ + sec θ) (tan θ – sec θ + 1)}/(tan θ – sec θ + 1)

= tan θ + sec θ

= (sin θ/cos θ) + (1/cos θ)

= (sin θ + 1)/cos θ

= (1 + sin θ)/cos θ = R.H.S. Proved.

FAQs about Trigonometric Formulas and Identities

What is Trigonometry?

Trigonometry is a branch of mathematics that focuses on the relationships between the angles and sides of triangles, particularly right-angled triangles.

What are three basic trigonometric ratios?

• Sin A = Perpendicular/ Hypotenuse
• Cos A= Base/Hypotenuse
• Tan A= Perpendicular/ Base

Which triangle are trigonometric formulas applicable to?

Trigonometric Formulas are applicable to Right Angled Triangles.

What are the major trigonometric ratios?

Sine, Cosine, Tangent, Cotangent, Secant and Cosecant.

For which angle is the value of the tan ratio equal to the cot ratio?

For the value of 45°, tan 45°= cot 45° = 1.

What is formula for sin3x?

Formula for sin3x is 3sin x – 4 sin³x.