Trigonometry formulas:- Trigonometry is a branch of mathematics that deal with angles, lengths and heights of triangles and relations between different parts of circles and other geometrical figures.
Maths Formulas – Trigonometric Ratios and identities are very useful and learning the below formulae help in solving the problems better. Trigonometry formulas are essential for solving questions in Trigonometry Ratios and Identities in Competitive Exams.
Trigonometric Identities are equalities that involve trigonometric functions and are true for every value of the occurring variables where both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles.
Trigonometric Ratio relationship between the measurement of the angles and the length of the side of the right triangle. These formulas relate lengths and areas of particular circles or triangles. On the next page you’ll find identities. The identities don’t refer to particular geometric figures but hold for all angles.
Trigonometry Formulas List
When we learn about trigonometric formulas, we consider it for right-angled triangles only. In a right-angled triangle, we have 3 sides namely – Hypotenuse, Opposite side (Perpendicular), and Adjacent side (Height). The longest side is known as the hypotenuse, the side opposite to the angle is perpendicular and the side where both hypotenuse and opposite side rests is the adjacent side.
Here is the list of formulas for trigonometry.
Sum and Difference Identities
Double Angle Identities
Triple Angle Identities
Half Angle Identities
Sum to Product Identities
Inverse Trigonometry Formulas
Formulas for arcs and sectors of circles
You can easily find both the length of an arc and the area of a sector for an angle θ in a circle of radius r.
Length of an arc. The length of the arc is just the radius r times the angle θ where the angle is measured in radians. To convert from degrees to radians, multiply the number of degrees by π/180.
Arc = rθ.
Trigonometric Formulas – Right Angle
The most important formulas for trigonometry are those for a right triangle. If θ is one of the acute angles in a triangle, then the sine of theta is the ratio of the opposite side to the hypotenuse, the cosine is the ratio of the adjacent side to the hypotenuse, and the tangent is the ratio of the opposite side to the adjacent side
Pythagorean theorem, the well-known geometric theorem that the sum of the squares on the legs of a right triangle is equal to the square on the hypotenuse (the side opposite the right angle)—or, in familiar algebraic notation, (P)2 + (B)2 = (H)2
Applying Pythagoras theorem for the given right-angled theorem, we have:
(Perpendicular)2 + (Base)2 = (Hypotenuse)2
⇒ (P)2 + (B)2 = (H)2
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 or identities are derived:
sin θ = Perpendicular/Hypotenuse
cos θ = Base/Hypotenuse
tan θ = Perpendicular/Base
sec θ = Hypotenuse/Base
cosec θ = Hypotenuse/Perpendicular
cot θ = Base/Perpendicular
Magical Hexagon for Trigonometry Identities
tan(x) = sin(x) / cos(x)
sin(x) = cos(x) / cot(x)
cos(x) = cot(x) / csc(x)
cot(x) = csc(x) / sec(x)
csc(x) = sec(x) / tan(x)
sec(x) = tan(x) / sin(x)
Counter clock Wise:
cos(x) = sin(x) / tan(x)
sin(x) = tan(x) / sec(x)
tan(x) = sec(x) / csc(x)
sec(x) = csc(x) / cot(x)
csc(x) = cot(x) / cos(x)
cot(x) = cos(x) / sin(x)
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. With the length and base side of the right triangle given, 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.
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.
Signs of Trigonometric Ratios
Co-function Identities (in Degrees)
The co-function or periodic identities can also be represented in degrees as:
- sin(π/2−θ) = cosθ
- cos(π/2−θ) = sinθ
- tan(π/2−θ) = cotθ
- cot(π/2−θ) = tanθ
- sec(π/2−θ) = cosecθ
- cosec(π/2−θ) = secθ
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]
Values of Trigonometric Functions
Some Important Books for Competitive Exam:
Q1: What trigonometry formulas should I study for the SSC CHSL?
Ans: For SSC CHSL, you should study trigonometry formulas either from your Class 10 textbook or from this article.
Q2: What are the basic trigonometric ratios?
Ans: Sine, Cosine, Tangent, Cotangent, Secant and Cosecant.
Q3: What are all the formulas of trigonometry?
Ans: You can learn all the trigonometry formulas from this article. You will get to know about:
– Reciprocal Relationship Between Trigonometric Functions
– Trigonometric Ratios Of Complementary Angles
– Periodicity Identities
– Trigonometric Identities
– Signs Of Trigonometric Functions
– Trigonometric Functions Of Sum And Difference Of Two Angles
– Trigonometry Formulas Involving Product Identities
– Trigonometry Formulas Involving Sum To Product Identities
– Trigonometry Formulas Involving Double Angle Identities
– Trigonometry Formulas Involving Triple Angle Identities
– Trigonometry Formulas Involving Half Angle Identities
– Trigonometry Formulas: Inverse Properties
Q4: What are formulas for trigonometry ratios?
Ans: Sin A = Perpendicular/Hypotenuse
Cos A = Base/Hypotenuse
Tan A = Perpendicular/Base
Q5: How do I memorize maths trigonometry formulas?
Ans: Our academic experts advise you not to memorize these trigonometry formulas. The more you try to learn consciously, the more is the chance that you are going to forget it. The best way to learn these formulas is to write them on a piece of paper and refer to them while you solve the questions. This way you will be able to easily learn the Trigonometry formulas.
Q6: What are the three main functions in trigonometry?
Ans: Sin, Cos and Tan are three main function in trigonometry.
Q7: Can I get a trigonometry formulas list?
Ans: Yes, with the help of this article, you can get all the important trigonometry formulas in one place.
Q8: What are the fundamental trigonometry identities?
Ans: The three fundamental identities are:
1. sin^2 A + cos^2 A = 1
2. 1+tan^2 A = sec^2 A
3. 1+cot^2 A = csc^2 A
Q9: Trigonometry formulas are applicable to which triangle?
Ans: Right-angled triangle
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