Trigonometry Formulas: Overview, Questions, Preparation

Trigonometric formulae are equations that formulate the relation between the lengths of the sides and angles of a right-angled triangle. It enables us to calculate the distances and angles in a Pythagorean landscape using basic real functions.

The Trigonometric formulae have been essential tools underlying the study of complex architecture, engineering, and numerous spatial sciences. These simple ratios are easy to learn and integrate with a general understanding of a right-angled triangle’s properties. 

Various Trigonometric formulae are used in contemporary Mathematics. These formulae are defined in terms of the different angles and sides of a Right-angled Triangle. They outline the relationship between the Hypotenuse, the base, and the perpendicular of a given Right-angled Triangle.

They are majorly classified into:

1. Trigonometric Ratios
2. Trigonometric Identities

From the earlier set of fundamental equations, we observe that some of the ratios are simply reciprocal.

The important Reciprocal Identities are:

Secant 

sec θ = 1/  sin θ

Cosecant

cos θ = 1/ cos θ

Cotangent

cot θ =  1/ tan θ

Trigonometric identities are defined as being periodic. This means that they repeat their values after a specified period. These equations hold good for the values of the variables and are substituted for different variables in problems. 

Basic Trigonometric Identities

1. sin^2 A + cos^2 A = 1
2. 1+tan^2 A = sec^2 A
3. 1+cot^2 A = cosec^2 A

Sum and Difference Identities

These formulae are used to simplify complex equations to simple additive or subtractive functions.

• 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)

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

First, we can study simple trigonometry formulas, especially about the right triangles, which have an angle θ, a hypotenuse, a side opposite angle θ, and a side adjacent to angle θ. 

Trigonometric ratios

The ratios of the trigonometry for the right-sided triangle can be written as: 

sinθ = Opposite side/Hypotenuse

cosθ = Adjacent Side/Hypotenuse

tanθ = Opposite side/Adjacent Side

secθ = Hypotenuse/Adjacent side

cosecθ = Hypotenuse/Opposite side

cotθ = Adjacent side/Opposite side

Calculus for a unit circle 

Related to the unit circle, the circle that has a circumference equal to one and for which θ is the angle. Equating the hypotenuse and one of the hands of a unit circle is the radius of the unit circle. 

Hypotenuse = Adjacent side to θ = 1

Therefore, trigonometry ratios are given by: 

• sin θ = y/1 = y
• cos θ = x/1 = x
• tan θ = y/x
• cot θ = x/y
• sec θ = 1/x
• cosec θ = 1/y

Trigonometric identities

Tangent and Cotangent Identities

tanθ = sinθcosθ

cotθ = cosθsinθ

Reciprocal Identities

sinθ = 1/cosecθ

cosecθ = 1/sinθ

cosθ = 1/secθ

secθ = 1/cosθ

tanθ = 1/cotθ

cotθ = 1/tanθ

Pythagorean Identities

sin2θ + cos2θ = 1

1 + tan2θ = sec2θ

1 + cot2θ = cosec2θ

The even and odd angles formulas

sin(-θ) = -sinθ

cos(-θ) = cosθ

tan(-θ) = -tanθ

cot(-θ) = -cotθ

sec(-θ) = secθ

cosec(-θ) = -cosecθ

Co-function Formulas

sin(90-θ) = cosθ

cos(90-θ) = sinθ

tan(90-θ) = cotθ

cot(90-θ) = tanθ

sec(90-θ) = cosecθ

cosec(90-θ) = secθ

Students get to learn trigonometric ratios and basic formulae and identity based on them in class 10th. In class 11th, formulae involving half-angle, double angle, triple angle identities and sum, difference, and product identities are introduced. The problems get eventually complicated, and in-class 12th, the syllabus covers inverse trigonometric functions. Nevertheless, the entire spectrum of formulae learnt over these three years will remain integral to solving the class XII board examination problems. Especially calculus occupies a significant part of the Class XII syllabus and demands the trigonometry formulae’s skilled application.

Trigonometry Formulas list in Class 10

This concept is taught under the chapter Introduction to Trigonometry. You will learn the trigonometric identities and use them to solve complex questions. The weightage of this chapter is 12 marks.

Trigonometry Formulas list in Class 11

Learning and memorizing these formulas can allow students in Classes 10, 11, and 12 to score good marks in trigonometry. The weightage of the unit is 23 marks.

Trigonometry Formulas list in Class 12

You can find a list of trigonometric functions under the chapter Inverse Trigonometric Functions under the unit Relations and Functions which are of 10 marks.

• Evaluate sin 18°/cos 72°      

Let’s convert sine function into cos function by using the formula, sinx = cos(90°-x)

So, sin 18° = cos(90°-18°)

=> sin 18° = cos 72°

Putting the value in the question,

=> cos 70°/cos 72° = 1

• Find the value of sin 765° 

Sin 765°  can be written as sin(2x365°+45° )

=sin(2π+45°) 

= sin 45°   { sin(2π+x) = x} 

= 1/√2 

• Find the value of sin 765° 

Sin 765°  can be written as sin(2x365°+45° )

=sin(2π+45°) 

= sin 45°   { sin(2π+x) = x} 

= 1/√2 

• What is the cos3x formula?

The formula for cos3x is cos3x = 4cos^3x - 3cosx (where x represents the angle)

• What is the cos2t formula?

The formula for Cos 2t = Cos2t – Sin2t. (where t is the angle of the right triangle)

• What is the formula of sin2x?

The formula for sin 2x is 2 sin x cos x. (where x is the angle of the right triangle)

Q: What are the standard formulas for trigonometry ratios?

Q: What are the fundamental trigonometric identities?

Q: Why are a few trigonometric values for standard angles not defined?

Q: What is the difference between geometry and trigonometry?

Q: How do I identify the sides of a right-angled triangle?

Q: What are the six trigonometric identities?

Q: Who is the father of trigonometry?

Q: What is sin(x) in mathematical formulas?

Q: What is the cosine?