Trigonometry is a branch of mathematics used to solve various numbers of problems using different formulae. The usage of trigonometry in real life is humungous. Some of the major places where trigonometry can be beneficial are: calculating the distance between the moon and earth and measuring the distance between two objects placed far from one another with missing inputs.
In this article, the students of CBSE Class 12 and JEE Main aspirants will be able to understand the different trigonometric formulae and identities while learning about their related concepts.
What is Trigonometry?
Trigonometry is a branch of Mathematics capable of dealing with a set of relations of the angles and sides of triangles. They also include their relevant functions at different angles. Trigonometry in JEE Main includes solving problems using different trigonometric formulae.
These problems include trigonometric ratios such as sin, sec, cos, cosec, tan and cot functions. Apart from that, they also consist of product identities, Pythagorean identities, etc.
Additionally, other formulae, including ratio signs in different quadrants, containing co-function identities, which are nothing but shifting angles, half-angle identities, double angle identities, sum and difference identities, etc., are explained in brief.
List of Trigonometric formulae
It is common to consider only right-angled triangles when it comes to the formulae. There are three sides in right-angled triangles as Adjacent side (base), the Opposite side (Perpendicular), and the hypotenuse (longest side).
The side present opposite the angle of the triangle is called the opposite side of the triangle. On the other hand, the hypotenuse is considered the longest side of the triangle and is often seen opposite the triangle’s right angle.
Given-below is the list of formulas that are being used in trigonometry:
- Basic trigonometric function formula
- Reciprocal identities
- Trigonometry table
- Cofunction identities
- Periodic identities
- Half angle identities
- Double angle identities
- Triple angle identities
- Sum and difference identities
- Sum to product identities
- Product identities
- Inverse trigonometry formulae
Basic trigonometric function formula
In CBSE Class 12 Mathematics, there are six ratios for finding the elements used in trigonometry. They are cosine, sine, tangent, cosecant, secant and cotangent. These are termed trigonometric functions.
With reference to the right-angled triangle, the following is the derivation of functions and identities used in trigonometry:
- sin = Opposite sideHypotenuse side
- cos = Adjacent sideHypotenuse side
- tan = Opposite sideAdjacent side
- sec = Hypotenuse sideAdjacent side
- cosec θ = Hypotenuse sideOpposite side
- cot cot = Adjacent sideOpposite side
Reciprocal identities
Reciprocal identities are nothing but the reciprocals of basic trigonometric formulae. They are:
- = 1cosec θ
- = 1sec θ
- = 1
- cosec θ = 1
- sec θ = 1cos θ
- cot θ = 1
For the JEE Main exam, the students are advised to be thorough with all the formulae and reciprocals. All these are usually taken from a right-angled triangle. When the base side and height are given, the students might be asked to find the values of cosine, sine, tangent, etc., using trigonometric formulae.
Trigonometry table
In CBSE Mathematics, there are some commonly used angles, for which the students might be asked to solve problems using them. The following is the tabulation of the same:
| 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 | ∞ |
| cosec | ∞ | 2 | √2 | 2/√3 | 1 | ∞ | -1 | ∞ |
| sec | 1 | 2/√3 | √2 | 2 | ∞ | -1 | ∞ | 1 |
Periodicity identities (in radians)
In Class 12, the periodic identities are the formulas which are helpful in shifting the angles by 2, π, 2 π, etc. They are also named as co-function identities.
- sin (2 – A) = A , and cos 2 – A = A
- sin (2 + A) = A , and cos 2 + A = – A
- sin (3 π2 + A) = – A , and cos 3 π2 + A = A
- sin (3 π2 – A) = – cos A , and cos 3 π2 – A = A
- sin π + A = – A , and cos π + A = – A
- sin π – A = A , and cos π – A = – A
- sin 2 π + A = A , and cos 2 π + A = A
- sin 2 π – A = – A , and cos 2 π – A = A
Usually, the nature of trigonometric identities is cyclic. These are capable of repeating themselves after the periodicity constant. They differ from one trigonometric identity to another. For example, 45 ° = 225 ° is different, whereas the value is true for 45 ° and cos 225 °. To verify the value, refer to the above table.
Cofunction identities (in degrees)
The periodic identities or the co-function identities are also illustrated in degrees as follows:
- sin 90 ° – x = x
- cos 90 ° – x = x
- tan 90 ° – x = x
- sec 90 ° – x = cosec x
- cosec 90 ° – x = x
- cot (90 ° – x) = x
Sum and difference identities
The following is the list of identities used for sum and difference in a trigonometric function:
- sin x + y = sin x cos y + cos 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
- cos x – y = cos x cos y + sin x sin y
- tan x + y = tan x + tan y 1 – tan x ∙ tan y
- tan x – y = tan x – tan y 1 + tan x ∙ tan y
Double angle identities
- sin 2 x = 2 sin x ∙ cos x = 2 tan x 1 + x
- cos 2 x = x – x = 1 – x 1 + x
- cos 2 x = 2 x – 1 = 1 – 2 x
- tan 2 x = 2tan x 1 – x
- sec (2 x) = x 2 – x
- cosec 2 x = sec x ∙ cosec x2
Triple angle identities
- sin 3 x = 3 sin x – 4 x
- cos 3 x = 4 x – 3 cos x
- tan 3 x = 3 tan x – x 1 – 3 x
Half angle identities
- x2 = ± 1 – cos x2
- x2 = ± 1 + cos x2
- x2 = 1 – cos (x)1 + cos (x)
Also,
x2 = 1 – cos x 1 + cos (x)
= 1 – cos x (1 – cos x) 1 + cos x (1 – cos x)
= (1 – cos x) 21 – (x)
= (1 – cos x) 2sin2 (x)
= 1 – cos (x)) sin x
So,
tan x2 = 1 – cos x sin x
Product identities
- x ∙ y = sin x + y + sin (x – y) 2
- x ∙ y = cos x + y + cos (x – y) 2
- sin x ∙ sin y = cos x – y – cos (x + y) 2
Sum to product identities
- x + sin y = 2 x + y2 x – y2
- x – y = 2 x + y2 x – y2
- x + y = 2 x + y2 x – y2
- x – y = – 2 x + y2 x – y2
Inverse trigonometry formulae
- (- x) = – x
- – x = π – x
- (- x) = – x
- – x = – x
- – x = π – x
- – x = π – x
Sin 3 x Formula
The students of JEE Main should know the sin 3 x formula when it comes to trigonometry. It can be described as the sine of three-time (3 *) of an angle present in a right-angled triangle. It can be expressed as:
sin 3 x = 3 sin x – 4 x
Major systems of trigonometry formulae
All the trigonometric formulae can be divided into two major systems:
- Trigonometric identities
- Trigonometric ratios
Trigonometric identities: They are nothing but the formulae which contain trigonometric functions. These trigonometric identities can be true for all the values of the variables.
Trigonometric ratio: They can be defined as the relationship between the length of the sides and the measurements of the angles of the triangle.
Watch our video to know the concepts on Trignometry.
FAQs
1. What are some of the real-time applications of trigonometry?
The usage of trigonometry in this world is tremendous. Following are some of them:
1. Physics uses trigonometry while solving real-time problems
2. The usage of trigonometry can be seen in video games
3. It is used while constructing buildings
4. It has a major application in aerospace engineering
5. Very much useful in measuring the height of the buildings and mountains
2. Is trigonometry important for the JEE Mains exam?
It is a known fact that trigonometry is considered one of the most important topics when it comes to JEE Mains. It is specified that at least 3 or 4 questions are asked from this particular chapter. So, the students are advised to learn it without fail.
3. Is there trigonometry in Class 12?
The Class 12 Maths consists of inverse trigonometric functions. This chapter is inclusive of graphs, definitions, and elementary properties of those inverse functions. The formulae in this chapter play a vital role during the time of examination.
4. What are the three basic formulas of trigonometry?
The following are the three fundamental formulas used in trigonometry:
1. Sine function: sin = Opposite sideHypotenuse side
2. Cosine function: cos = Adjacent sideHypotenuse side
3. Tangent function: tan = Opposite sideAdjacent side
5. What does in mathematics?
Can be pronounced as theta. Theta is used commonly as a variable in a trigonometric equation which helps indicate angles.
Conclusion
All the CBSE Class 12 students are advised to practice the trigonometric concepts well to be easier for them to appear for board exams and JEE Main. The students should practice all the formulae and identities in trigonometry to gain confidence while appearing for the examinations.
Therefore, in this article, we have provided a clear idea of trigonometry, along with its list of formulae and identities. The students were also taught trigonometric tables, which are key to understanding and learning all the other related concepts.
