Trigonometry formulas

If you are looking for the complete list of all trigonometry formulas for classes 10, 11, and 12, we have made it completely easy for you to understand and learn all trigonometry formulas and ratios to a single page. Memorizing these trigonometric formulas help students to solve trigonometry problems easily, which further leads to a good score in Mathematics. For easy understanding and access, we have provided the trigonometry table and inverse trigonometry formulas.

Trigonometry, a word derived from the Greek words ‘Trigon’ and ‘Metron’ refers to ‘measuring the sides of a triangle. This branch of Mathematics studies the relation between angles and sides of triangles. Trigonometry formulas are  used widely in various fields and have been in use since the 3rd Century BC. From Navigation to Celestial Mechanics to Engineering, Architecture and other various fields trigonometry play a vital role. 

Various geometrical problems are solved using the trigonometric ratios Sine (Sin), Cosine (Cos), Tangent (Tan), Cotangent (cot), Secant (sec), and Cosecant (Csc), product identities, Pythagorean Identities, etc. There are several Trigonometric functions, formulas, and ratios like the sign of ratios in different quadrants, sum identities, difference identities, cofunction identities, double angle identities, half-angle identities, etc.  These are introduced from Class 10th and the concept is further explained in classes 11th & 12th.

Trigonometry Formulas for Class 10, 11, and 12:

As trigonometry is the study of relationships between angles and sides of the triangles. The primary triangle studied is the right triangle. In a right-angled triangle, the three sides are named Hypotenuse, Opposite, and Adjacent. The longest side is called the Hypotenuse, the opposite side to the angle is called the Perpendicular (Opposite), and the side where both hypotenuse and opposite sits is the adjacent side.

• Basic Formulas
• Reciprocal Identities
• Trigonometry Table
• Periodic Identities
• Co-function Identities
• Sum and Difference Identities
• Double Angle Identities
• Triple Angle Identities
• Half Angle Identities
• Product Identities
• Sum to Product Identities
• Inverse Trigonometry Formulas

 

Basic Formulae

The basic Trigonometry formulas for class 10 are six basic ratios which are used for finding the elements in trigonometry which are called trigonometric functions. These are Sine (Sin), Cosine (Cos), Tangent (Tan), Cotangent (Cot), Secant (Sec), and Cosecant (Csc).

By taking right triangle as a reference, these Trigonometry functions are derived as following:

• 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

Reciprocal identities are given as following:

• 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

These are taken from a right-angled triangle. As the height and base of the right triangle are given, we can find out the Sine (sin), Cosine (cos), Tangent (tan), Cotangent (cot), Secant (sec), and Cosecant (CSC) values using trigonometric formulas. These reciprocal trigonometric identities can also be derived by using the trigonometric functions.

Trigonometry Table

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

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	∞
csc	∞	2	√2	2/√3	1	∞	-1	∞
sec	1	2/√3	√2	2	∞	-1	∞	1

Periodicity Identities in Radian

After the basic formulas Trigonometry formula for Class 11 introduces periodicity Identities used in shifting the trigonometric functions by one period to the left or right. These are 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

We know tall trigonometry functions in Class 11 are repetitive in nature, repeating themselves after this periodicity constant. This periodicity constant is different for different trigonometric identities. tan 45° = tan 225° but this is also true for cos 45° and cos 225°.

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]

Trigonometry formulas for class 10, 11 and 12

 

List of Full Essential Trigonometry Formulas for Class 10, 11, and 12 

Trigonometry Formulas 

Based on Trigonometry ratios like Sine, Cosine, Tangent, Cotangent, Secant, and Cosecant. All the important trigonometry formulas will adhere here that will help to solve the complex trigonometry problems. Practically trigonometry is the study of triangles. It is the most important trigonometry formula for the students of classes 10,11 and 12. This important trigonometry formula has been formulated based on the right-angled triangle and relies on identities and ratios. The trigonometry formulas also help to analyze the pivotal relationship between angles and the length of the sides of a triangle. 

With the reference to Unit Circle and Right-angled triangle, the study of trigonometry formulas will be elaborated. 

Let focus on pivotal Trignometry Formulas that are important for the students of Class 10,11 and 12 considering the approach of the Right-angle Triangle that has  θ angle, a hypotenuse, the side opposite angle to angle θ, and the side adjacent to angle θ.

Trigonometry Ratio for Right-Angled Triangle 

The basic trigonometry ratio considering the approach of Right-Angled Triangle  can be elaborated as 

Unit Circle Trignometry Ratio 

In Unit Circle the value of radius is equal to 1 and θ is the only angle. whereas, the 

Value of the hypotenuse and adjacent side is always equal to the radius in the unit circle. 

Hypotenuse = Adjacent side to θ = 1

Therefore, trigonometry ratio for Unit Circle can be written as

Important Trigonometry Identities  (Pdf Download)

Download Tirgonometry Identities

Unit Circle 

Unit Circle assists us to analyze the several values of cos and sin
ratios of numerous angles like 0°,30°,45°,60°,90° in the four quadrants 

Source 

Frequently Asked Questions – FAQs

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

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

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

The three fundamental identities are:
sin^2 A + cos^2 A = 1
1+tan^2 A = sec^2 A
1+cot^2 A = csc^2 A

Right-angled triangle