In Mathematics, trigonometry is one of the most fundamentally used concepts. The usage of trigonometry is crucial in real life apart from the textbook. The JEE Main 2022 aspirants are motivated by their tutors to study this chapter well to obtain the best possible marks. This chapter might seem hard, but once the students are used to the formulae and identities, it will be a simple task.
In this article, we will be talking about trigonometry in general. We will have a detailed understanding of all the related concepts and formulae and revision notes for students to secure maximum marks.
|Table of contents|
|Trigonometry – Definition|
|Basic trigonometric ratios|
|Derived trigonometric ratios|
|Basic trigonometric identities|
|Trigonometric identities – Compound angles|
|Trigonometric identities – Multiple angles|
|Applications of trigonometry|
Trigonometry – Definition
Trigonometry is a term that comes from a Greek word, which means measurement. In simple words, it can be described as the term used in measuring right-angled triangles. There are certain numbers of trigonometric ratios which can be obtained from the sides of the triangle.
Multiple trigonometric ratios can be easily formed by utilising both the length and angle of the right-angled triangles. Theta is the angle between two sides and can be represented as. The following are the basic trigonometric ratio forms:
Basic trigonometric ratios
Sine, abbreviated as sin, is a basic trigonometric function. It can be defined as the ratio of the opposite side of the right-angled triangle to its hypotenuse side. Its derivation is,
sin = Opposite sideHypotenuse side
The second basic form of a trigonometric function is cosine, which can be represented as cos. The sine and cosine functions are considered the fundamentals which are capable of deriving the rest of the basic trigonometric functions such as secant, cosecant, tangent and cotangent.
Cosine is the ratio of the right-angled triangle’s base length to its hypotenuse side. It can be represented as mentioned below:
cos = Adjacent sideHypotenuse side
The derivation of the trigonometric function tangent is from the other basic trigonometric functions such as sine and cosine. The abbreviation of this is tan and is expressed as the ratio between sine and cosine.
Tan can be described as the opposite side of the right-angled triangle to its adjacent side. The following is the mathematical representation of the same:
tan = Opposite sideAdjacent side
In JEE Main, all these sine, cosine and tangent functions are referred to as basic trigonometric ratios.
Derived trigonometric ratios
Derived trigonometric functions are nothing but the ratios which are derived from the basic functions by inverting them. Therefore, the aspirants of JEE should be aware of this topic as well.
Cosecant, also called cosec, can be obtained from sine. It is nothing but sine’s multiplicative inverse. Cosec is written as the ratio between the hypotenuse side and the opposite side of the right-angled triangle.
cosec θ = Hypotenuse sideOpposite side
While inversing the basic trigonometric function, cosine, another function is formed and is termed as secant. Secant can be abbreviated as sec. It is defined as the ratio between the hypotenuse side and the adjacent side of the right-angled triangle.
sec = Hypotenuse sideAdjacent side
Cotangent can be obtained using the multiplicative inverse of the tangent. A cotangent can be represented as a cot. It is nothing but the ratio of the adjacent side to the opposite side of the right-angled triangle.
cot = Adjacent sideOpposite side
To learn trigonometry, the JEE 2022 aspirants should be able to know the ratios at standard angles. So, the students need to have the table stored in their memory. Apart from mathematical functions, this table is also helpful for solving Chemistry and Physics questions. The students are advised to follow it precisely.
The below-mentioned is the table for 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|
Basic trigonometric identities
The basic trigonometric identities are usually asked indirectly in the JEE Main examinations. The problems can sometimes be complex in nature. However, a regular practice on this topic would be beneficial in remembering during the time of examination. The following are the identities derived from basic trigonometric ratios:
- sin 2 (a) +cos 2 (a) = 1
- 1 + cot 2 a = cosec 2 a
- 1 +tan 2 a =sec 2 a
- cosec a – cot a = 1 cosec a + cot a
- sin 4 (a) +cos 2 (a) = 1 – 2 cos 2 a sin 2 (a)
- sec (a) – tan a = 1 sec a + tan (a)
Trigonometric identities – Compound angles
Consider an angle which is the sum or difference of the other two angles, whose values are unknown. In this case, to calculate the trigonometric ratio value, the aspirants must memorise the following identities, that consist of compound angles. It is used to solve any question simply. The below-mentioned are the standard trigonometric identities:
- sin (K ± L) =sin K cos L ± cos K sin L
- cos K ± L =cos K cos L ∓ sin K sin L
- tan K + L =tan K ± tan L 1 ± tan K ∙ tan L
- cot K + L =cot K ∙ cot L F ∓ 1 cot L ± cot K
- sin K + L sin K – L =sin 2 K – sin 2 L = cos 2 L – cos 2 K
- cos K + L cos (K – L) = cos 2 K – sin 2 L =cos 2 L – sin 2 K
- sin (K + L + M) = sin K cos L cos M + sin L cos K cos M +sin M cos K cos L – sin K sin L sin M
- cos K + L + M =cos K cos L cos M – cos K sin L sin M – cos L sin K sin M -cos M sin K sin L
- tan (K + L + M) =tan K + tan L +tan M – tan K tan L tan M 1 -tan K ∙ tan L – tan L ∙ tan M -tan M ∙ tan K
Trigonometric identities – Multiple angles
In the JEE 2022 exam, there are a few questions which can be asked based on multiple angles. The following are the formulae:
- sin 2 K = 2 sin K cos K = 2K 1 + tan 2 K
- sin 3 K = 3 sin K – 4 sin 3 K
- cos 2 K = 2cos 2 K – 1 = 1 – 2 sin 2 K =(1 – tan 2 K)/(1 + tan 2 K)=cos 2 K – sin 2 K
- cos 3 K = 4cos 3 K – 3cos K
- tan 2 K =2 tan K 1 – 2 K
- tan 3 K = 3 tan K – tan 3 K 1 – 3 tan 2 K
- sin M +sin N = 2 sin (M + N 2) cos (M – N 2)
- sin M – sin N = 2 cos (M + N 2) sin (M – N 2)
- 2 sin L cos K = sin (K + L) – sin (K – L)
- 2 cos K cos L=cos (K + L) + cos (K – L)
- 2 sin K sin L = cos (K – L) – cos (K + L)
- 2 cos K cos L =cos K + L + cos K – L
- cos M – cos N = 2 cos (M + N 2) cos (M – N 2)
- cos M – cos N = 2 sin (M + N 2) sin (N – M 2)
In trigonometric equations, n belongs to Z. Here Z means integers. The following the tabulation for the same:
|sin (a) = 0||n π|
|cos (a) = 0||2 n + 1 π 2|
|tan (a) = 0||n π|
|sin (a) = sin (b)||n π + – 1 n ∙ b|
|cos a =cos (b)||2 n π ± b|
|tan (a) = tan (b)||n π + b|
Applications of trigonometry
Trigonometry is being used in so many different fields in this world. Its real-world applications are enormous. The following are some of them:
- The usage of trigonometry can be seen in criminology
- Marine biology acts as a great source in the usage of trigonometry
- In the aviation sector, trigonometry plays a major role
- Navigation needs basic trigonometry concepts to perform
- It is very much helpful in measuring the height of mountains and buildings
To conclude, since it is a fundamental concept, the students are taught trigonometry’s definition and its basic and derived trigonometric ratios. Moreover, a deep understanding of the trigonometric table is also being taught.
In addition to this, basic trigonometric identities, compound angles and multiple angles were also discussed. Lastly, transformation formulae and trigonometric equations are discussed concerning the formulae.
1. What is the main idea of trigonometry?
Trigonometry is nothing but a branch of Maths, which helps study the relationships between the angles and sides of the right-angled triangles. The usage of trigonometry can be seen hugely in geometry. It is because all the straight-side shapes are easily broken into separate pieces of triangles.
2. What is the hypotenuse side and how can you find its value?
Hypotenuse side is usually the longest side of the right-angled triangle. To find it, there is a hypotenuse formula, which can be used. This hypotenuse formula is derived from the Pythagoras theorem. It can be expressed as,
Hypotenuse2 = Altitude2 + Base2
Hypotenuse = a2 + b2
3. Who is the founder of trigonometry?
A Greek mathematician named Hipparchus introduced the first known table of chords in 140 BC. Although it has been found that his tables haven’t survived, there was a huge claim that he wrote at least twelve books on chord tables. Due to this, Hipparchus is called the father of trigonometry.
4. How can trigonometry be used in music?
In the music field, trigonometry can be used in measuring the pitch or level of a musical note, which is nothing but the sound waves. These notes are calculated in hertz. Sound engineers use studios, which are panels contained in rooms. It turns the waves in specific angles, which will make the waves bounce off in an appropriate manner which ends up making a song even and smooth.
5. What are the two types of trigonometry?
Trigonometry can be classified into two major types. They are:
- Plane trigonometry
- Spherical trigonometry