Trigonometry Formulas

Trigonometry formulas are sets of different formulas involving trigonometric identities, used to solve problems based on the sides and angles of a right-angled triangle. These trigonometry formulas include trigonometric functions like sine, cosine, tangent, cosecant, secant, cotangent for given angles. Let us learn these formulas involving Pythagorean identities, product identities, co-function identities (shifting angles), sum & difference identities, double angle identities, half-angle identities, etc. in detail in the following sections.

1.	List of Trigonometry Formulas
2.	Basic Trigonometry Formulas
3.	Trigonometry Formulas Involving Reciprocal Identities
4.	Trigonometric Ratio Table
5.	Trigonometry Formulas Involving Periodic Identities(in Radians)
6.	Trigonometry Formulas Involving Co-function Identities(in Degrees)
7.	Trigonometry Formulas Involving Sum and Difference Identities
8.	Multiple and Sub-Multiple Angles
9.	Sum and Product of Identities
10.	Inverse Trigonometry Formulas
11.	Trigonometry Formulas Involving Sine and Cosine Laws
12.	FAQs on Trigonometry Formulas

Trigonometry formulas can be classified into different categories based on the trigonometry identities involved. Let us look at the below sets of different trigonometry formulas.

• Basic Trig Ratio Formulas: These are trigonometry formulas relating to the basic trigonometric ratios sin, cos, tan, etc.
• Reciprocal Identities: This includes trigonometry formulas dealing with the reciprocal relationship between trig ratios.
• Trigonometric Ratio Table: Trigonometry values are depicted for standard angles in the trigonometry table.
• Periodic Identities: These comprise trigonometry formulas that help in finding values of trig functions for a shift in angles by π/2, π, 2π, etc.
• Co-function Identities: Trigonometry formulas for cofunction identities depict interrelationships between the trigonometry functions.
• Sum and Difference Identities: These trigonometry formulas are used to find the value of the trigonometry function for the sum or difference in angles.
• Half, Double and Triple Identities: These trigonometry formulas include values of trig functions for half, double or triple angles.
• Sum to Product Identities: These trigonometry formulas are used to represent the product of trigonometry functions as their sum or vice-versa.
• Inverse Trigonometry Formulas: Inverse trigonometry formulas include the formulas related to inverse trig functions like sine inverse, cosine inverse, etc.
• Sine Law and Cosine Law

Some basic trigonometry formulas can be observed in the image below. Let us study them in detail in the following sections.

Basic trigonometry formulas are used to find the relationship of trig ratios and the ratio of the corresponding sides of a right-angled triangle. There are basic 6 trigonometric ratios used in trigonometry, also called trigonometric functions- sine, cosine, secant, co-secant, tangent, and co-tangent, written as sin, cos, sec, csc, tan, cot in short. The trigonometric functions and identities are derived using a right-angled triangle as the reference. We can find out the sine, cosine, tangent, secant, cosecant, and cotangent values, given the dimensions of a right-angled triangle, using trigonometry formulas as,

Trigonometric Ratio Formulas

• sin θ = Perpendicular/Hypotenuse
• cos θ = Base/Hypotenuse
• tan θ = Perpendicular/Base
• sec θ = Hypotenuse/Base
• cosec θ = Hypotenuse/Perpendicular
• cot θ = Base/Perpendicular

Cosecant, secant, and cotangent are the reciprocals of the basic trigonometric ratios sine, cosine, and tangent. All of the reciprocal identities are also derived using a right-angled triangle as a reference. These reciprocal trigonometric identities are derived using the trigonometric functions. The trigonometry formulas on reciprocal identities, given below, are used frequently to simplify trigonometric problems.

• 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

Here is a table for trigonometry formulas for angles that are commonly used for solving trigonometry problems. The trigonometric ratios table helps in finding the values of trigonometric standard angles such as 0°, 30°, 45°, 60°, and 90°.

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 formulas involving periodic identities are used to shift the angles by π/2, π, 2π, etc. All trigonometric identities are cyclic in nature which means that they repeat themselves after a period. This period differs for different trigonometry formulas on periodic identities. For example, tan 30° = tan 210° but the same is not true for cos 30° and cos 210°. You can refer to the trigonometry formulas given below to verify the periodicity of sine and cosine functions.

First Quadrant:

• sin (π/2 – θ) = cos θ
• cos (π/2 – θ) = sin θ
• sin (π/2 + θ) = cos θ
• cos (π/2 + θ) = – sin θ

Second Quadrant:

• sin (3π/2 – θ) = – cos θ
• cos (3π/2 – θ) = – sin θ
• sin (3π/2 + θ) = – cos θ
• cos (3π/2 + θ) = sin θ

Third Quadrant:

• sin (π – θ) = sin θ
• cos (π – θ) = – cos θ
• sin (π + θ) = – sin θ
• cos (π + θ) = – cos θ

Fourth Quadrant:

• sin (2π – θ) = – sin θ
• cos (2π – θ) = cos θ
• sin (2π + θ) = sin θ
• cos (2π + θ) = cos θ

The trigonometry formulas on cofunction identities provide the interrelationship between the different trigonometry functions. The co-function trigonometry formulas are represented in degrees below:

• 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

The sum and difference identities include the trigonometry formulas of sin(x + y), cos(x - y), cot(x + y), etc.

• 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) = (tan x + tan y)/(1 - tan x • tan y)
• 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) = (tan x - tan y)/(1 + tan x • tan y)

Trigonometry formulas for multiple and sub-multiple angles can be used to calculate the value of trigonometric functions for half angle, double angle, triple angle, etc.

Trigonometry Formulas Involving Half-Angle Identities

The half of the angle x is presented through the below few trigonometry formulas.

sin (x/2) = ±√[(1 - cos x)/2]

cos (x/2) = ± √[(1 + cos x)/2]

tan (x/2) = ±√[(1 - cos x)/(1 + cos x)]

or, tan (x/2) = ±√[(1 - cos x)(1 - cos x)/(1 + cos x)(1 - cos x)]

tan (x/2) = ±√[(1 - cos x)²/(1 - cos²x)]

⇒ tan (x/2) = (1 - cos x)/sin x

Trigonometry Formulas Involving Double Angle Identities

The double of the angle x is presented through the below few trigonometry formulas.

• sin (2x) = 2sin(x) • cos(x) = [2tan x/(1 + tan² x)]
• cos (2x) = cos²(x) - sin²(x) = [(1 - tan² x)/(1 + tan² x)]
• cos (2x) = 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

Trigonometry Formulas Involving Triple Angle Identities

The triple of the angle x is presented through the below few trigonometry formulas.

• sin 3x = 3sin x - 4sin³x
• cos 3x = 4cos³x - 3cos x
• tan 3x = [3tanx - tan³x]/[1 - 3tan²x]

Trigonometric formulas for sum or product identities are used to represent the sum of any two trigonometric functions in their product form, or vice-versa.

Trigonometry Formulas Involving Product Identities

• 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 Involving Sum to Product Identities

The combination of two acute angles A and B can be presented through the trigonometric ratios, in the below trigonometry formulas.

• 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)]

Using the inverse trigonometry formulas, trigonometric ratios are inverted to create the inverse trigonometric functions, like, sin θ = x and θ = sin ⁻¹x. Here x can have values in whole numbers, decimals, fractions, and exponents.

• 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

Sine Law: The sine law and the cosine law give a relationship between the sides and angles of a triangle. The sine law gives the ratio of the sides and the angle opposite to the side. As an example, the ratio is taken for the side 'a' and its opposite angle 'A'.

(sin A)/a = (sin B)/b = (sin C)/c

Cosine Law: The cosine law helps to find the length of aside, for the given lengths of the other two sides and the included angle. As an example the length 'a' can be found with the help of the other two sides 'b' and 'c' and their included angle 'A'.

• a² = b² + c² - 2bc cosA
• b² = a² + c² - 2ac cosB
• c² = a² + b² - 2ab cosC

where, a, b, c are the lengths of the sides of the triangle, and A, B, C are the angles of the triangle.

Related Topics

• Sin Cos Tan
• Trigonometric Identities
• Cosecant Secant and Cotangent Functions
• Trigonometric Ratios
• Trigonometry Formulas For Class 10
• Trigonometry Calculator

 

What are Trigonometry Formulas?

Trigonometry formulas are used to solve problems based on the sides and angles of a right-angled triangle, using the different trigonometric identities. These formulas can be used to evaluate trigonometric ratios(also referred to as trigonometric functions), sin, cos, tan, csc, sec, and cot.

What is the Basic Trigonometry Formula?

Basic trigonometry formulas involve the representing of basic trigonometric ratios in terms of the ratio of corresponding sides of a right-angled triangle. These are given as, sin θ = Opposite Side/Hypotenuse, cos θ = Adjacent Side/Hypotenuse, tan θ = Opposite Side/Adjacent Side.

What are Trigonometry Ratios' Formulas?

The three main functions in trigonometry are Sine, Cosine, and Tangent. Trigonometry ratios' formulas are given as,

• Sine Function: sin(θ) = Opposite / Hypotenuse
• Cosine Function: cos(θ) = Adjacent / Hypotenuse
• Tangent Function: tan(θ) = Opposite / Adjacent

What are Trigonometry Formulas for Even and Odd Identities?

The trigonometry formulas involving even and odd identities are given as,

• sin(–x) = –sin x
• cos(–x) = cos x
• tan(–x) = –tan x
• csc (–x) = –csc x
• sec (–x) = sec x
• cot (–x) = –cot x

What are the Trigonometry Formulas Involving Pythagorean Identities?

The three fundamental trigonometry formulas involving the Pythagorean identities are given as,

• sin²A + cos²A = 1
• 1 + tan²A = sec²A
• 1 + cot²A = cosec²A

Trigonometry Formulas Are Applicable to Which Triangle?

Trigonometry formulas are applicable to right-angled triangles. These trig formulas represent the trigonometric ratios in terms of the ratio of corresponding sides of a right-angled triangle.

What are Addition Trigonometry Formulas?

The trigonometry formulas for trigonometry ratios when the angles are in addition are given as,

• 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) = (tan x + tan y)/(1 - tan x • tan y)

How to Remember Trigonometry Formulas Easily?

The trick to learn basic trigonometry formulas is using the mnemonic "SOHCAHTOA", which can be used to memorize trigonometric ratios as,
SOH: Sine = Opposite / Hypotenuse
CAH: Cosine = Adjacent / Hypotenuse
TOA: Tangent = Opposite / Adjacent

What is sin 3x Trigonometry Formula?

Trigonometry formula, sin 3x is the sine of three times of an angle in a right-angled triangle, it is expressed as: sin 3x = 3sin x - 4sin³x.