✅ Trigonometric formulas ⭐️⭐️⭐️⭐️⭐

Basic Trigonometric Function Formulas

There are basically 6 ratios used for finding the elements in Trigonometry. They are called trigonometric functions. The six trigonometric functions are sine, cosine, secant, co-secant, tangent and co-tangent.

By using a right-angled triangle as a reference, the trigonometric functions and identities are derived:

• sin θ = Opposite Side/Hypotenuse
• cos θ = Adjacent Side/Hypotenuse
• tan θ = Opposite Side/Adjacent Side
• sec θ = Hypotenuse/Adjacent Side
• cosec θ = Hypotenuse/Opposite Side
• cot θ = Adjacent Side/Opposite Side

Reciprocal Identities

The Reciprocal Identities are given as:

• 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 these are taken from a right angled triangle. When height and base side of the right triangle are known, we can find out the sine, cosine, tangent, secant, cosecant, and cotangent values using trigonometric formulas. The reciprocal trigonometric identities are also derived by using the trigonometric functions.

Pythagorean Identities

Trigonometry Table

Below is the table for trigonometry formulas for angles that are commonly used for solving problems.

Periodicity Identities (in Radians)

These formulas are used to shift the angles by π/2, π, 2π, etc. They are also called 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 repeat themselves after this periodicity constant. This periodicity constant is different for different trigonometric identities. tan 45° = tan 225° but this is true for cos 45° and cos 225°. Refer to the above trigonometry table to verify the values.

Co-function Identities (in Degrees)

The co-function or periodic identities can also be represented in degrees as:

• sin(90°−x) = cos x
• cos(90°−x) = sin x
• tan(90°−x) = cot x
• cot(90°−x) = tan x
• sec(90°−x) = csc x
• csc(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) = (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)

Double Angle Identities

• 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)
• csc (2x) = (sec x. csc x)/2

Triple Angle Identities

• Sin 3x = 3sin x – 4sin³x
• Cos 3x = 4cos³x-3cos x
• Tan 3x = [3tanx-tan³x]/[1-3tan²x]

Half Angle Identities

Product identities

Sum to Product Identities

Inverse Trigonometry Formulas

What is Sin 3x Formula?

Sin 3x is the sine of three times of an angle in a right-angled triangle, that is expressed as:

Sin 3x = 3sin x – 4sin³x

Sine and Cosine Laws

Sine laws: The sine law and the cosine law give us the relationship between the sides and the angles of a triangle. The Sine Law gives the ratio of the sides and the angle opposite to the side. 

Example: The ratio can be taken for the side a and its opposite angle ‘A’.

Cosine laws: The cosine Law helps to find the length of a side for the given lengths of the other two sides and the included angle.  For example, the length ‘a′ can be found with the help of sides b and c, and their included angle A.

Basic Trigonometric Formulas

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Trigonometry Formulas From Class 10 to Class 12

Trigonometry Formulas For Class 10

Applying Pythagoras theorem for the given right-angled triangle, we have:

(Perpendicular)²+(Base)²=(Hypotenuse)²

⇒(P)²+(B)²=(H)²

The Trigonometric formulas are given below:

Reciprocal Relation Between Trigonometric Ratios

Trigonometric Ratios Table

The below trigonometry table formula shows trigonometry formulas and commonly used angles for solving trigonometric problems. The trigonometric ratios table helps find the values of standard trigonometric angles like 0∘,30∘,45∘,60∘0∘,30∘,45∘,60∘ and 90∘.

Trigonometric Sign Functions

• sin (-θ) = − sin θ
• cos (−θ) = cos θ
• tan (−θ) = − tan θ
• cosec (−θ) = − cosec θ
• sec (−θ) = sec θ
• cot (−θ) = − cot θ

Trigonometric Identities

1. sin²A + cos²A = 1
2. tan²A + 1 = sec²A
3. cot²A + 1 = cosec²A

Periodic Identities

• sin(2nπ + θ ) = sin θ
• cos(2nπ + θ ) = cos θ
• tan(2nπ + θ ) = tan θ
• cot(2nπ + θ ) = cot θ
• sec(2nπ + θ ) = sec θ
• cosec(2nπ + θ ) = cosec θ

Periodicity formulas or identities are utilised to shift the angles by π/2,π, and 2π The periodicity identities are also termed the co-function identities. All the trigonometric identities are cyclic, which means they repeat themselves after a period. The period differs for various trigonometric identities.

Complementary Ratios

Quadrant I

• sin(π/2−θ) = cos θ
• cos(π/2−θ) = sin θ
• tan(π/2−θ) = cot θ
• cot(π/2−θ) = tan θ
• sec(π/2−θ) = cosec θ
• cosec(π/2−θ) = sec θ

Quadrant II

sin(π−θ) = sin θ
cos(π−θ) = -cos θ
tan(π−θ) = -tan θ
cot(π−θ) = – cot θ
sec(π−θ) = -sec θ
cosec(π−θ) = cosec θ

Quadrant III

• sin(π+ θ) = – sin θ
• cos(π+ θ) = – cos θ
• tan(π+ θ) = tan θ
• cot(π+ θ) = cot θ
• sec(π+ θ) = -sec θ
• cosec(π+ θ) = -cosec θ

Quadrant IV

• sin(2π− θ) = – sin θ
• cos(2π− θ) = cos θ
• tan(2π− θ) = – tan θ
• cot(2π− θ) = – cot θ
• sec(2π− θ) = sec θ
• cosec(2π− θ) = -cosec θ

Sum and Difference of Two Angles

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

Double Angle Formulas

• sin2A = 2sinA cosA = [2tan A + (1+tan²A)]
• cos2A = cos²A–sin²A = 1–2sin²A = 2cos²A–1= [(1-tan²A)/(1+tan²A)]
• tan 2A = (2 tan A)/(1-tan²A)

Thrice of Angle Formulas

• sin3A = 3sinA – 4sin³A
• cos3A = 4cos³A – 3cosA
• tan3A = [3tanA–tan³A]/[1−3tan²A]

Trigonometry Formulas For Class 11

Sign of Trigonometric Functions in Different Quadrants

Basic Trigonometric Formulas for Class 11

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

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

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

sin (A -B) = sin A cos B – cos A sin B

Based on the above addition formulas for sin and cos, we get the following below formulas:

• sin(π/2-A) = cos A
• cos(π/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

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

• tan(A+B) = [(tan A + tan B)/(1 – tan A tan B)]
• tan(A-B) = [(tan A – tan B)/(1 + tan A tan B)]

If none of the angles A, B and (A ± B) is a multiple of π, then

• cot(A+B) = [(cot A cot B − 1)/(cot B + cot A)]
• cot(A-B) = [(cot A cot B + 1)/(cot B – cot A)]

Some additional formulas for sum and product of angles:

• cos(A+B) cos(A–B)=cos²A–sin²B=cos²B–sin²A
• sin(A+B) sin(A–B) = sin²A–sin²B=cos²B–cos²A
• sinA+sinB = 2 sin (A+B)/2 cos (A-B)/2

Formulas for twice of the angles:

• sin2A = 2sinA cosA = [2tan A /(1+tan²A)]
• cos2A = cos²A–sin²A = 1–2sin²A = 2cos²A–1= [(1-tan²A)/(1+tan²A)]
• tan 2A = (2 tan A)/(1-tan²A)

Formulas for thrice of the angles:

• sin3A = 3sinA – 4sin³A
• cos3A = 4cos³A – 3cosA
• tan3A = [3tanA–tan³A]/[1−3tan²A]

Trigonometry Formulas For Class 12

Basic Concepts

Here are the domain and range of basic trigonometric functions:

• Sine function, sine: R → [– 1, 1]
• Cosine function, cos : R → [– 1, 1]
• Tangent function, tan : R – { x : x = (2n + 1) π/2, n ∈ Z} →R
• Cotangent function, cot : R – { x : x = nπ, n ∈ Z} →R
• Secant function, sec : R – { x : x = (2n + 1) π/2, n ∈ Z} →R – (– 1, 1)
• Cosecant function, cosec : R – { x : x = nπ, n ∈ Z} →R – (– 1, 1)

Properties of Inverse Trigonometric Functions

• sin^-1 (1/a) = cosec^-1(a), a ≥ 1 or a ≤ – 1
• cos^-1(1/a) = sec^-1(a), a ≥ 1 or a ≤ – 1
• tan^-1(1/a) = cot^-1(a), a>0
• sin^-1(–a) = – sin^-1(a), a ∈ [– 1, 1]
• tan^-1(–a) = – tan^-1(a), a ∈ R
• cosec^-1(–a) = –cosec^-1(a), | a | ≥ 1
• cos^-1(–a) = π – cos^-1(a), a ∈ [– 1, 1]
• sec^-1(–a) = π – sec^-1(a), | a | ≥ 1
• cot^-1(–a) = π – cot^-1(a), a ∈ R

Addition Properties of Inverse Trigonometry functions

• sin^-1a + cos^-1a = π/2, a ∈ [– 1, 1]
• tan^-1a + cot^-1a = π/2, a ∈ R
• cosec^-1a + sec^-1a = π/2, | a | ≥ 1
• tan^-1a + tan^-1 b = tan^-1 [(a+b)/1-ab], ab<1
• tan^-1a – tan^-1 b = tan^-1 [(a-b)/1+ab], ab>-1
• tan^-1a – tan^-1 b = π + tan^-1[(a+b)/1-ab], ab > 1; a,b > 0

Twice of Inverse of Tan Function

• 2tan^-1a = sin^-1 [2a/(1+a²)], |a| ≤ 1
• 2tan^-1a = cos^-1[(1-a²)/(1+a²)], a ≥ 0
• 2tan^-1a = tan^-1[2a/(1+a²)], – 1 < a < 1

Trigonometry Formulas Major systems

All trigonometric formulas are divided into two major systems:

• Trigonometric Identities
• Trigonometric Ratios

Trigonometric Identities are formulas that involve Trigonometric functions. These identities are true for all values of the variables. Trigonometric Ratio is known for the relationship between the measurement of the angles and the length of the sides of the right triangle.

Here we provide a list of all Trigonometry formulas for the students. These formulas are helpful for the students in solving problems based on these formulas or any trigonometric application. Along with these, trigonometric identities help us to derive the trigonometric formulas, if they will appear in the examination.

We also provided the basic trigonometric table pdf that gives the relation of all trigonometric functions along with their standard values. These trigonometric formulae are helpful in determining the domain, range, and value of a compound trigonometric function. Students can refer to the formulas provided below or can also download the trigonometric formulas pdf that is provided above.

Solved Problems

Q.1:What is the value of (sin30° + cos30°) – (sin 60° + cos60°)?

Sol: Given,

(sin30° + cos30°) – (sin 60° + cos60°)

= ½ + √3/2 – √3/2 – ½

= 0

Q.2: If cos A = 4/5, then tan A = ?

Sol: Given, 

Cos A = ⅘

As we know, from trigonometry identities,

1+tan²A = sec²A

sec²A – 1 = tan²A

(1/cos²A) -1 = tan²A

Putting the value of cos A = ⅘.

(5/4)² – 1 = tan² A

tan²A = 9/16

tan A = 3/4

Frequently Asked Questions – FAQs

What are the basic trigonometric ratios?

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

What are formulas for trigonometry ratios?

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

What are the three main functions in trigonometry?

Sin, Cos and Tan are three main functions in trigonometry.

What are the fundamental trigonometry identities?

The three fundamental identities are:
1. sin² A + cos² A = 1
2. 1+tan² A = sec² A
3. 1+cot² A = csc² A

Trigonometry formulas are applicable to which triangle?

Right-angled triangle