All Trigonometry Formulas and Identities: Full Formula List From Basic To Advanced Maths

Trigonometry Formulas: There are very few topics in mathematics that trouble students more than trigonometry and calculus. In fact, it is the base of many advanced math concepts and is also utilized in other subjects like physics. As such, it’s paramount that students learn trigonometry by heart. There are hundreds of formulas and identities that students have to memorize during their school time.

Trigonometry and subsequently calculus are a headache for students but necessary to clear the exams and pursue higher education in mathematics. Trigonometry also has many real-world applications.

As the name suggests, trigonometry is the branch of mathematics that deals with the study of the relationship between sides and angles of a right triangle. It’s used in astronomy, cartography, geography, naval and aviation industries. We bring you the following trigonometry formulas pdf.

Also Read: Simple and Easy Tricks to Remember Trigonometric Formulas

Trigonometry Formulas PDF

Fundamentals of Trigonometry for Class 10

The trigonometric ratios of the angle A in right triangle ABC, given above are defined as follows:

• sine of ∠ A = side opposite to angle A/hypotenuse  = BC/AC
• cosine of ∠ A = side adjacent to angle A/hypotenuse = AB/AC
• tangent of ∠ A = side opposite to angle A/side adjacent to angle A = BC/AB
• cosecant of ∠ A = 1/sine of ∠A = AC/BC
• secant of ∠ A = 1/cosine of ∠A = AC/AB
• cotangent of ∠ A = 1/tangent of ∠A = AB/BC

Trigonometry Table

Trigonometric 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

Co-Function Identities 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

Trigonometry Formulas for Class 11

There are several trigonometric formulas and identities students have to learn in Class 11. These lay the foundation of other advanced concepts in calculus and without memorising them by heart, no student will be able to efficiently study trigonometry and it’s related fields.

Since we have already covered the basics like the Trigonometry Table and Triangle formulas, let’s jump straight to the identities. But first you should know what are radians and degree.

Radian

• The Radian is the SI unit used to measure angles and 1 radian is defined as the angle made at the center of a circle by an arc whose length is equal to the radius of the circle.
• Radian is used to calculate the angle in terms of radius. It’s denoted by “rad” or the symbol “c” in exponent.
• If an angle is written without any units, then it means that it is in radians.
• Radian formula is: arc length/radius

Degree

• A degree is also a unit used to denote the measurement of an angle. Degree is used mostly in practical geometry while radian is used in mathematics and physics.
• A degree can be defined as the angle made by one part of 360 equally divided parts of a circle at the centre with a radius of r.

Basic Trigonometry Formulas

sin(−θ) = −sin θ

cos(−θ) = cos θ

tan(−θ) = −tan θ

cosec(−θ) = −cosecθ

sec(−θ) = sec θ

cot(−θ) = −cot θ

Product to Sum Formulas

sin x sin y = 1/2 1/2[cos(x-y) - cos (x+y)]

cos x cos y = 1/2 1/2[cos(x-y) + cos (x+y)]

sin x cos y = 1/2[sin(x+y) + sin(x−y)]

cos x sin y = 1/2[sin(x+y) – sin(x−y)]

Sum to Product Formulas

sin x + sin y = 2 sin (x+y)/2 cos (x−y)/2

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

cos x + cos y = 2 cos (x+y)/2 cos (x−y)/2

cos x – cos y = -2 sin (x+y)/2 sin (x−y)/2

Basic Identities

Sin²x + cos²x = 1

1+tan²x = sec²x

1+cot²x = cosec²x

Sign of Trigonometric Functions in Different Quadrants

Basic Trigonometric Formulas for Class 11

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

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

sin (x+y) = sin x cos y + cos x sin y

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

If none of the angles x, y and (x ± y) is an odd multiple of π/2, then

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

If none of the angles x, y and (x ± y) is a multiple of π, then

• cot(x+y) = [(cot x cot y − 1)/(cot y + cot x)]
• cot(x-y) = [(cot x cot y + 1)/(cot y – cot x)]

Some additional formulas for sum and product of angles:

• cos(x+y) cos(x–y)= cos2x–sin2y=cos2y–sin2x
• sin(x+y) sin(x–y) = sin2x–sin2y=cos2y–cos2x
• sinx+siny= 2 sin (x+y)/2 cos (x-y)/2

Formulas for twice of the angles:

• sin2x= 2sinxcosx = [2tan x /(1+tan2x)]
• cos2x= cos2x–sin2x = 1–2sin2x = 2cos2x–1= [(1-tan2x)/(1+tan2x)]
• tan 2x= (2 tan x)/(1-tan2x)

Formulas for thrice of the angles:

• sin3x= 3sinx – 4sin3x
• cos3x= 4cos3x – 3cosx
• tan3x= [3tanx–tan3x]/[1−3tan2x]

Trigonometry Formulas for Class 12

Domain and Range of Trigonometric Functions

With the addition of Relation and Functions in Class 11 and 12, the concepts of domain and range also becomes applicable in Trigonometry. You must know the following Domain Range values of trigonometric functions to fully comprehend calculus in Class 12.

Inverse Trigonometric Functions

READ: CBSE Class 12 Maths Chapter 2 Inverse Trigonometric Functions Formulas List, Important Definitions & Examples

• sin^-1x = - sin^-1x
• cos^-1x = π - cos^-1x
• tan^-1(-x) = -tan^-1x
• cosec⁻¹(-x) = -cosec⁻¹x
• sec^-1(-x) = π - sec^-1x
• cot^-1(-x) = π - cot^-1x
• sin^-1x + cos^-1x = π/2
• tan^-1x + cot^-1x = π/2
• sec^-1x + cosec^-1x = π/2
• sin^-1x = - sin^-1x
• cos^-1x = π - cos^-1x
• tan^-1(-x) = -tan^-1x
• cosec⁻¹(-x) = -cosec⁻¹x
• sec^-1(-x) = π - sec^-1x
• cot^-1(-x) = π - cot^-1x
• sin^-1x + cos^-1x = π/2
• tan^-1x + cot^-1x = π/2
• sec^-1x + cosec^-1x = π/2

Double Of Inverse Trigonometric Functions

• 2tan^-1x = sin^-1(2x/1+ x²)
• = cos^-1(1-x²/1+x²)
• = tan^-1(2x/1-x²)
• 2sin^-1x = sin^-1(2x.√(1 - x²))
• 2cos^-1x = cos^-1(2x²- 1)

Triple Of Inverse Trigonometric Functions

• 3sin^-1x = sin^-1(3x - 4x³)
• 3cos^-1x = cos^-1(4x³- 3x)
• 3tan^-1x = tan^-1(3x - x³/1 - 3x²)

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