Trigonometry: Learn about Ratios, Sides, Angles, and Identities in detail!
Saurav Sinha
| Updated: May 6, 2021 11:19 IST
Trigonometry is a branch of mathematics that deals with the relationship between sides and angles connected through ratios. It moreover helps in the calculation of angles and sides of a triangle with the help of different trigonometric ratios. It is applied in different domains including those of engineering, architecture, physics, surveying, astronomy and so on. These are the topics that are frequently asked in all sorts of government exams like SSC JE, and SSC CGL, followed by banking exams like SBI PO, SBI Clerk, IBPS PO, IBPS Clerk, etc.
With the growing competition in the examination, the need for a strong concept and a clear idea of a subject is getting more and more important. One such important competitive exam topic of mathematics is trigonometry. By the end of the article, you will be well acknowledged with the different trigonometric ratios, identities, the standard angles followed by applications.
List of Topics under Trigonometry
In the introduction, we read about the trigonometry basics, let us start our discussion with a quick overview of the different topics that are counted under the subject.
| Related Topics | Definition |
| Trigonometric Identities | Trigonometric identities are useful whenever trigonometric functions are contained in an equation. The different identities are reciprocal, Pythagorean identities, opposite angle identities, complementary angle identities followed by angle sum and difference identities. |
| Trigonometric Ratios | Trigonometric ratios are the values of all the trigonometric functions depending on the value of the ratio of sides in a right-angled triangle. The six basic trigonometric ratios are sine, cosine, tangent, cosecant, secant, and cotangent. |
| Trigonometric Functions | Trigonometric functions are widely used in calculus, geometry, algebra and other related domains. The functions sin, cos and tan are the primary classifications of trig functions. |
| Inverse Trigonometric Functions | Inverse trigonometric functions as per the name are the inverse functions of the basic trigonometric functions. They are written as; arcsin(x), arccos(x), arctan(x), arccsc(x), arcsec(x)and arccot(x). |
| Derivatives of Trigonometric Function | Derivatives of trigonometric functions in mathematics is the approach to finding the derivative of a trigonometric function. These are obtained using the concepts of calculus. |
| Limits of Trigonometric Function | Limits of trigonometric functions deal with calculating the limits of the six different trigonometric functions. The limits are determined considering the continuity, domain and range of the functions. |
| Trigonometry Formulas | This topic involves a complete set of formulas ranging from sum and difference formulas, double angle formulas, half-angle formulas, product identities, etc. in detail. |
| Trigonometry Table | The trigonometric table helps us locate the different values of standard trigonometric angles; 0°, 30°, 45°, 60°, 90°, 180°, 270°, and 360°. Such a table is quite beneficial for quick revision. |
| Trigonometry Graphs | The trigonometric function graphs help us obtain the domain and range of a given function. Such graphs are used for modelling different electronics, and mechanical and scientific phenomena in engineering and science. The graphs of such functions are periodic in nature. |
If you have mastered Trigonometry, you can also learn about Sequences and Series in detail here!
Trigonometric Ratios
To understand the concept of trigonometric ratios let us consider the below figure:
The figure depicts a triangle ΔABC, right-angled at B and angle A = θ. The side which is opposite to θ is the perpendicular or opposite side. The side that is opposite to angle B is termed the hypotenuse and the side opposite to angle C is known as the base or adjacent side. Now let’s try to write the formula for the sin, cos, tan, etc.
Sine Function(sin)= perpendicular/ Hypotenuse=BC/AC
Cosine Function(cos)= base / Hypotenuse=AB/AC
Tangent Function(tan)=perpendicular / base=BC/AB
Cosecant Function(cosec)=Hypotenuse / perpendicular=AC/BC
Secant Function(sec)=Hypotenuse / base =AC/AB
Cotangent Function(cot)=base/ perpendicular=AB/BC
If you are reading Trigonometric theories, also read about Three Dimensional Geometry here.
Important Trigonometric Identities
The list of all the important trigonometric identities in mathematics is given below:
Fundamental Identities
The basic fundamental identities are as follows:
- sin²x + cos²x = 1
- 1 +tan²x= sec²x
- 1 + cot²x = cosec²x
Sum, Difference, and Products of Two Angles Identities
The various sum, difference, and products of two angles identities are listed below:
- sin (A ± B) = sin A cos B ± cos A sin B
- cos (A ± B) = cos A cos B ∓ sin A sin B
- 2 sin A cos B = sin (A + B) + sin (A – B)
- 2 cos A sin B = sin (A + B) – sin (A – B)
- 2 cos A cos B = cos (A + B) + cos (A – B)
- 2 sin A sin B = cos (A – B) – cos (A + B)
- \(\tan\left(a\ +b\right)=\frac{\left(\tan a\ +\tan b\right)}{1-\tan a\tan b}\)
- \(\tan\left(a\ -b\right)=\frac{\left(\tan a\ -\tan b\right)}{1+\tan a\tan b}\)
- \(\cot\left(a\ +b\right)=\frac{\left(\cot a\cot b-1\right)}{\cot b+\cot a}\)
- \(\cot\left(a\ -b\right)=\frac{\left(\cot a\cot b+1\right)}{\cot b-\cot a}\)
- \(\sin c+\sin d=2\sin\left(\frac{c+d}{2}\right)\cos\left(\frac{c-d}{2}\right)\)
- \(\sin c-\sin d=2\sin\left(\frac{c-d}{2}\right)\cos\left(\frac{c+d}{2}\right)\)
Multiple Angle Identities
The different multiple angle identities are listed below:
- sin(2A) = 2sin(A) cos(A) = [2tan A/(1+tan²A)]
- cos 2A = cos²A– sin²A= [(1-tan²A)/(1+tan²A)]
- tan 2A = 2 tan A / (1 – tan²A)
- cot 2A = (cot²A – 1) / 2 cot A
- cos(2A) = 2cos²A−1 = 1–2sin²A
- \(\tan3A=[3\tan A-\tan^3A]/[1-3\tan^2A]\)
- \(\cot3A=(3\cot x-Cot^3x)/(1-3Cot^2x)\)
- \(Sin3A=3\sin A{-4}\sin^3A\)
- \(\cos3A=4\cos^3A-3\cos A\)
Trigonometric Functions in Different Quadrants
Now that we know about the different trigonometric functions along with the related formulas, below is a summary of the sign of different functions in the different quadrants in a tabular format.
| Trigonometric Function | Quadrant I | Quadrant II | Quadrant III | Quadrant IV |
| Sin | + | + | – | – |
| Cos | + | – | – | + |
| Cosec | + | + | – | – |
| Sec | + | – | – | + |
| Tan | + | – | + | – |
| Cot | + | – | + | – |
Trigonometric Values of some Standard Angles
In the previous header, we read about signs of the six functions in different functions. The below trigonometry table shows the different values of the functions in terms of degree.
| Function | 0° | 30° | 45° | 60° | 90° |
| sin | 0 | 1 / 2 | 1 / √2 | √ 3 / 2 | 1 |
| cos | 1 | √ 3 / 2 | 1 / √2 | 1 / 2 | 0 |
| Tan | 0 | 1 / √3 | 1 | √3 | Undefined |
| Cot | Undefined | √3 | 1 | 1 / √3 | 0 |
| Sec | 1 | 2 / √3 | √2 | 2 | Undefined |
| cosec | Undefined | 2 | √2 | 2 / √3 | 1 |
Also, read about Relations and Functions here.
Domain and Range of Trigonometric Functions
The value of θ in sin, cos, tan, cosec, sec and cot denotes the domain of the trigonometric functions and the resultant of the value is the range of the trig functions. The domain and range of of different trigonometric functions are as follows:
| Trigonometric functions | Domain | Range |
| sin θ | R | [-1, 1] |
| cos θ | R | [-1, 1] |
| tan θ | \(R-\left\{\left(2n+1\right)\frac{\pi}{2}:n\ \in Z\right\}\) | R |
| cosec θ | R – {nπ : n ∈ Z} | R – (-1, 1) |
| sec θ | \(R-\left\{\left(2n+1\right)\frac{\pi}{2}:n\ \in Z\right\}\) | R – (-1, 1) |
| cot θ | R – {nπ : n ∈ Z} | R |
Trigonometric Equation with their General Solution
An equation that involves one or more trigonometric ratios of unknown angles is termed a trigonometric equation. The solution to such an equation with variable x, such that it lies in between 0≤x≤2π is said to be a principal solution. On the other hand, if the solution holds an integer say ‘n’ in it, then the equation is called a general solution. The trigonometric equation with the respective general solution of different functions are listed below:
| Trigonometric Equation | General Solution |
| sin θ = 0 | θ = nπ, n ∈ Z |
| cos θ = 0 | θ = (2n + 1)π / 2, n ∈ Z |
| tan θ = 0 | θ = nπ, n ∈ Z |
| sin θ = 1 | θ = 2nπ + π / 2, n ∈ Z |
| cos θ = 1 | θ = 2nπ, n ∈ Z |
| sin θ = sin α | \(θ=n\pi+\left(-1\right)^nα\), α ∈ (-π/2,π/2), n ∈ Z |
| cos θ = cos α | θ = 2nπ ± α, α ∈ [0, π], n ∈ Z |
| tan θ = tan α | θ = nπ + α, α ∈ (-π/2,π/2) , n ∈ Z |
Also, read about General Equation of a Line here.
Trigonometric Functions Graphs
Properties like domain, range, periodicity and nature of the different functions can be best studied by employing the trigonometric function graphs. Below is the graph for sin, cos, tan, cosec, sec, and cot.
Also, read about Types of Functions here.
Applications of Trigonometry
Until now, we have read about the basic concepts of trigonometry with the formula, identities, angles and more. Trigonometry is also used in physics, engineering, satellite navigation, electronics study, architecture, astronomy, oceanography and many such domains. The different real-life applications are listed below:
Real-Life Examples of Trigonometry
- Let us understand the different applications of mathematical trigonometry in various domains.
- In physics subject, the concepts of trigonometry and its formula are used in vectors; to obtain the dot product and cross products, to obtain the components of the vector and so on.
- The trigonometry concepts are even involved in the navigation to point toward a location. The reason is that trigonometric functions can be used to set directions such as east-west or north-south.
- Further, the idea of trigonometry is also used in estimating the height of a building or a mountain, as this can be easily determined using the different trigonometric functions.
- It is also used in the aviation and naval industries.
- Trigonometric functions are also applied to determine the trajectory of a projectile and further estimate the causes of a collision. Also, the concept is applied to locate how an entity falls or at what angle a bullet is shot from a gun. Hence it is used in criminology as well.
- It is also used in marine biology, to measure the depth of sunlight to study the effects on algae that survive on photosynthesis.
| If you are checking the Trigonometry article, also check the related maths articles in the table below: | |
| Statistics | Integral Calculus |
| Partnership | Percentages |
| Rhombus | Circles |
Trigonometry FAQs
Q.1 What is the trigonometry meaning in maths?
Ans.1 Trigonometry in mathematics deals with the relationship between sides and angles connected through ratios. It helps calculate the angles and sides of a triangle with the help of different trigonometric ratios.
Q.2 What are the standard angles used in trigonometry?
Ans.2 0°, 30°, 45°, 60°, 90°, 180°, 270°, and 360° are some of the standard angles used in trigonometry.
Q.3 What are the six basic trigonometric functions?
Ans.3 The 6 basic trigonometric functions are as follows: sin, cos, tan, sec, cot, and cosec function.
Q.4 What are the primary function of trigonometry?
Ans.4 There are three primary functions of trigonometry; sine, cos, and tan function.
Q.5 What are the applications of trigonometry?
Ans.5 One of the prominent real-life applications of trigonometry is in the computation of height and distance followed by navigation, aviation department, marine biology, criminology, creation of maps, satellite systems, and more.
