Trigonometric Functions

One of the most important concepts in mathematics is trigonometry. The measuring of angles and sides of a triangle is the subject of trigonometry. Normally, trigonometry is used to solve problems with right-angled triangles. Its functions are also used to calculate the length of a circle's arc, which is a portion of a circle with a radius and a centre point. Trigonometry has applications across Physics, Engineering, Architecture, etc. This article is about the concept of trigonometric functions class 10, and trigonometric functions class 11.

Trigonometry, in its simplest form, is the study of triangles with angles and lengths on their sides. When we break down the word trigonometry, we find that 'Tri' is a Greek word that means three,' 'Gon' means length,' and metry' means trigonometric formulas measurement.'

What are Trigonometric Functions?

In trigonometry, there are six basic trigonometric functions. These functions are trigonometric ratios that are based on ratios of sides in a right triangle: the hypotenuse (the longest side), the base (the side adjacent to a chosen angle), and the perpendicular (the side opposite the chosen angle). These functions are sine, cosine, tangent, secant, cosecant, and cotangent. They help us find different values in triangles by comparing these side lengths.

Trigonometric Functions Formulas

The trigonometric functions formulas are

1. Sine ($\sin$)- $\sin$ is defined as a ratio of the side opposite to that angle (perpendicular) and hypotenuse.

$\sin t=\frac{\text { opposite }}{\text { hypotenuse }}$

2. Cosine ($\cos$) - $\cos$ is defined as the ratio of the side adjacent to that angle (base) and hypotenuse.

$\cos t=\frac{\text { adjacent }}{\text { hypotenuse }}$

3. Tangent ($\tan$) - $\tan$ is defined as the ratio of the side opposite to that angle (perpendicular) and the side adjacent to that angle (base).

$\tan t=\frac{\text { opposite }}{\text { adjacent }}$

4. cotangent ($\cot$) - $\cot$ is defined as the ratio of the side adjacent to that angle (base) and the side opposite to that angle (perpendicular). It is the reciprocal of tan.

$\cot t=\frac{\text { adjacent }}{\text { opposite }}$

5. secant ($\sec$) - $\sec$ is defined as the ratio of the hypotenuse and the side adjacent to that angle (base). It is reciprocal of cos.

$\sec t=\frac{\text {hypotenuse }}{\text { adjacent }}$

6. Cosecant ($\csc$) - $\csc$ is defined as the ratio of the hypotenuse and the side opposite to that angle (perpendicular). It is reciprocal of the sin.

$\csc t=\frac{\text { hypotenuse }}{\text { opposite }}$

Values of Trigonometric Functions

In the trigonometric functions table, we use the values of trigonometric ratios for standard angles $0°, 30°, 45°, 60°,$ and $90º$. It is easy to use the table as a reference to calculate values of trigonometric ratios for various other angles, using the trigonometric ratio formulas for existing patterns within trigonometric ratios and even between angles.

Below are the trigonometric ratios of some special angles $0°, 30°, 45°, 60°,$ and $90º$

Domain and Range of Trigonometric Functions

The domain and range of trigonometric functions,

1. Sine ($\sin$)

Domain is $R$

Range is $[-1,1]$

2. Cosine ($\cos$)

Domain is $R$

Range is $[-1,1]$

3. Tangent ($\tan$)

Domain is $\mathbb{R}-\left\{\frac{(2 \mathrm{n}+1) \pi}{2}, \mathrm{n} \in \mathbb{I}\right\}$
Range is $R$

4. Cosecant ($\csc$)

Domain is $\mathrm{R}-\{\mathrm{n} \pi, \mathrm{n} \in \mathrm{I}$ (Integers) $\}$
Range is $R-(-1,1)$

5. Secant ($\sec$)

Domain is $\mathbb{R}-\left\{\frac{(2 \mathrm{n}+1) \pi}{2}, \mathrm{n} \in \mathbb{I}\right\}$
The range is $R-(-1,1)$

6. Cotangent ($\cot$)

Domain is $\mathrm{R}-\{\mathrm{n} \pi, \mathrm{n} \in \mathrm{I}$ (Integers) $\}$
Range is $R$

Graph of Trigonometric Functions

The graph of trigonometric functions are

1. Sine ($\sin$)

$y = f(x) = \sin x$

Domain is $R$
Range is $[-1,1]$
We observe that $\sin x$ completes one full cycle of its possible values (from $-1$ to $1$ )
in the interval of length $2 \pi$.
So, the period of $\sin x$ is $2 \pi$.
The graph drawn in the interval $[0,2 \pi]$ repeats to the right and the left.
$\operatorname{Sin} x$ passes the graph as $\sin x=0$
So, The root of $y=\sin x$ is $n \pi$ where

$
n=\ldots \ldots-3,-2,-1,0,1,2,3,4 \ldots \ldots
$

Min value of $\sin x=-1$ at $=3 \pi / 2$
The max value of $\sin x=1$ at $\pi / 2$
Amplitude $=1$

$
\sin (-x)=-\sin x
$
So, $\sin \mathrm{x}$ is an odd function.

2. Cosine ($\cos$)

$y = f(x) = \cos x$

$
\sin (x+90)=\cos x
$

So, the $\cos x$ graph is drawn by shifting the $\sin \mathrm{x}$ graph by $90$ degrees.
Domain is $R$
Range is $[-1,1] \cos x=\pi / 2$. So, The root of $y=\cos x$ is $n(\pi / 2)$ where

$
n=\ldots \ldots-3,-2,-1,1,2,3,4 \ldots \ldots
$

Min value of $\cos x=-1$ at $2 \pi$
The max value of $\cos x=1$ at $0$ and $4 \pi$
Amplitude $=1$
Line of Symmetry $=\mathrm{Y}$ - axis

$
\cos (-x)=\cos x
$

So, $\cos x$ is an even function

3. Tangent ($\tan$)

$y=f(x) = \tan x$

Domain is $\mathbb{R}-\left\{\frac{(2 \mathrm{n}+1) \pi}{2}, \mathrm{n} \in \mathbb{I}\right\}$
Range is $R$
We observe that $\tan x$ increase in each of the intervals $\ldots$ $(-3 \pi / 2,-\pi / 2),(-\pi / 2, \pi / 2),(\pi / 2,3 \pi / 2) \ldots \ldots$.

The graph goes from negative to positive infinity. Period $=180$ degree $=\pi$. The root of the function $y=\tan x$ is $n \pi$ where $\mathrm{n}=\ldots \ldots-3,-2,-1,1,2,3 \ldots \ldots$

The amplitude is undefined as the graph tends to infinity.
Line of symmetry $=$ origin

$
\tan (-x)=-\tan x
$

So, the tangent function is odd.

4. Cosecant($\csc$)

$ y= f(x) = \csc x$

Domain is $\mathrm{R}-\{\mathrm{n} \pi, \mathrm{n} \in \mathrm{I}$ (Integers) $\}$
Range is $R-(-1,1)$
As the curve repeats after an interval of $2 \pi$, the period of the cosecant function is $2 \pi$.
The amplitude of the graph of a cosecant function is undefined as the curve does not have a maximum or a minimum value and tends to infinity.
Line of symmetry= origin
Vertical asymptotes are $x=n \pi$

$
\operatorname{cosec}(-x)=-\operatorname{cosec} x
$

So, $\operatorname{cosec} x$ is an odd function.

5. Secant ($\sec$)

$y=f(x)= \sec x$

Domain is $\mathbb{R}-\left\{\frac{(2 \mathrm{n}+1) \pi}{2}, \mathbf{n} \in \mathbb{I}\right\}$
The range is $R-(-1,1)$
The amplitude of the graph of a secant function is undefined as the curve does not have a maximum or a minimum value and tends to infinity. As the curve repeats after an interval of $2 \pi$, the period of the sunction is $2 \pi$.
Line of Symmetry $=\mathrm{Y}$-axis
Vertical asymptotes $=\mathrm{x}=(2 \mathrm{n}+1) \pi / 2$
$\operatorname{Sec}(-x)=\sec x$
So, it is an even function

6. Cotangent ($\cot$)

$y=f(x)= \cot x$

Domain is $\mathrm{R}-\{\mathrm{n} \pi, \mathrm{n} \in \mathrm{I}$ (Integers) $\}$
Range is $R$
The graph goes from negative to positive infinity.nWe observe that $\cot x$ decreases in each of the intervals $\ldots(-2 \pi /,-\pi),(-\pi, 0),(0, \pi) \quad$
The root of the equation $y=\cot x$ is $n \pi / 2$ where $n=\quad-3,-2,-1,1,2,3 . \quad$
The amplitude is undefined as the graph tends to infinity.

As the curve repeats after an interval of $\pi$, the period of the cotangent function is $\pi$. Line of symmetry $=$ origin
Vertical asymptotes are $x=n \pi$.

$
\cot (-x)=-\cot x
$

So it is an odd function.

Trigonometric Functions Identities

In trigonometry, there are numerous identities that help in solving a variety of problems. These formulas are essential for tackling complex trigonometric questions efficiently. They provide fundamental tools that enable quick solutions to various trigonometric problems. Let's explore the essential trigonometric identities used in these calculations.

1. Pythagorean theorem identities

The trigonometric identities that follow Pythagoras's theorem are called Pythagorean theorem identities.

$\begin{aligned} & \sin ^2 t+\cos ^2 t=1 \\ & 1+\tan ^2 t=\sec ^2 t \\ & 1+\cot ^2 t=\csc ^2 t \\ & \tan t=\frac{\sin t}{\cos t}, \quad \cot t=\frac{\cos t}{\sin t}\end{aligned}$

2. Reciprocal identities

Trigonometric ratios which are reciprocal to other trigonometric ratios are called reciprocal identities.

The reciprocal trigonometric identities are

$\begin{aligned} & \text { Cosecant } \quad \csc t=\frac{\text { hypotenuse }}{\text { opposite }}=\frac{1}{\sin t} \\ & \text { Secent } \quad \sec t=\frac{\text { hypotenuse }}{\text { adjacent }}=\frac{1}{\cos t} \\ & \text { Cotangent } \quad \cot t=\frac{\text { adjacent }}{\text { opposite }}=\frac{1}{\tan t}\end{aligned}$

3. Addition of angles trignometric identites

There are three trigonometric identities are related to the Addition of angles.

If $A$ and $B$ are two angles.
a) $\sin (A+B)=\sin A \cos B+\cos A \sin B$
b) $\cos (A+B)=\cos A \cos B-\sin A \sin B$
c) $\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A \tan B}$

4. Difference of angles trigonometric identities

There are three trigonometric identities related to the difference in angles.

If $A$ and $B$ are two angles.
a) $\sin (A-B)=\sin A \cos B-C \cos A \sin B$
b) $\operatorname{Cos}(A-B)=\cos A \cos B-\operatorname{Sin} A \sin B$
c) $\tan (A-B)=\frac{\tan A-\tan B}{1+\tan A \tan B}$

5. Double angle trignometric identities

The double angle trigonometric identities is used to transform the trigonometric ratios of double angles into the trignometric ratios of single angles.

If A is the angle of the triangle,
a) $\sin 2 A=2 \sin A \cos A$
b) $\cos 2 \mathrm{~A}=\cos ^2 A-\sin ^2 A=2 \cos ^2 A-1=1-2 \sin ^2 A$
c) $\tan 2 A=\frac{2 \tan A}{1-\tan ^2 A}$

6. Trigonometric identities of allied angles

Two angles are called allied if their sum or difference is a multiple of $\pi / 2(90)$
Below are some trigonometric identities of allied angles

- $\sin (90-\theta)=\cos (\theta)$
- $\cos (90-\theta)=\sin (\theta)$
- $\tan (90-\theta)=\cot (\theta)$
- $\csc (90-\theta)=\sec (\theta)$
- $\sec (90-\theta)=\csc (\theta)$
- $\cot (90-\theta)=\tan (\theta)$
- $\sin (90+\theta)=\cos (\theta)$
- $\cos (90+\theta)=-\sin (\theta)$
- $\tan (90+\theta)=-\cot (\theta)$
- $\csc (90+\theta)=\sec (\theta)$
- $\sec (90+\theta)=-\csc (\theta)$
- $\cot (90+\theta)=-\tan (\theta)$

7. Triple-angle Trigonometric identities

If angles are triple then trigonometric identities of $\sin \mathrm{A}, \cos \mathrm{A}$, and $\tan \mathrm{A}$ are

If A is the angle of the triangle,

$\begin{aligned} & \sin 3 \mathrm{~A}=3 \sin \mathrm{A}-4 \sin ^3 \mathrm{~A} \\ & \cos 3 A=4 \cos ^3 A-3 \cos A \\ & \tan 3 A=\frac{3 \tan A-\tan ^3 A}{1-3 \tan ^2 A} \\ & \cot 3 \mathrm{~A}=\frac{\cot ^3 \mathrm{~A}-3 \cot \mathrm{A}}{3 \cot ^2 \mathrm{~A}-1}\end{aligned}$

8. Trigonometric identities of supplementary angles

Two angles are called supplementary angles if their sum is 180.
Below are the trigonometric identities of supplementary angles:

- $\sin (180-\theta)=\sin (\theta)$
- $\cos (180-\theta)=-\cos (\theta)$
- $\tan (180-\theta)=-\tan (\theta)$
- $\csc (180-\theta)=\csc (\theta)$
- $\sec (180-\theta)=-\sec (\theta)$
- $\cot (180-\theta)=-\cot (\theta)$

9. Sum-product trigonometric identities

The Sum/Difference identity is used to express the sum or difference of sine and cosine functions into the Product of sine and cosine functions. Below are some Product sum trigonometric identities :

$\begin{aligned} & \sin \mathrm{C}+\sin \mathrm{D}=2 \sin \frac{\mathrm{C}+\mathrm{D}}{2} \cos \frac{\mathrm{C}-\mathrm{D}}{2} \\ & \sin \mathrm{C}-\sin \mathrm{D}=2 \cos \frac{\mathrm{C}+\mathrm{D}}{2} \sin \frac{\mathrm{C}-\mathrm{D}}{2} \\ & \cos \mathrm{C}+\cos \mathrm{D}=2 \cos \frac{\mathrm{C}+\mathrm{D}}{2} \cos \frac{\mathrm{C}-\mathrm{D}}{2} \\ & \cos \mathrm{C}-\cos \mathrm{D}=-2 \sin \frac{\mathrm{C}+\mathrm{D}}{2} \sin \frac{\mathrm{C}-\mathrm{D}}{2}\end{aligned}$

10. Trigonometric identities of the opposite angle

Below are some trigonometric identities of opposite angles:

$\begin{aligned}
& \operatorname{Sin}(-A)=-\sin A \\
& \operatorname{Cos}(-A)=\cos A \\
& \operatorname{Tan}(-A)=-\operatorname{Tan} A \\
& \operatorname{Cot}(-A)=-\cot A \\
& \operatorname{Sec}(-A)=\sec A\\
&\operatorname{Cosec}(-A)=-\operatorname{cosec} A
\end{aligned}$

Inverse Trigonometric Functions

In order to use inverse trigonometric functions, we need to understand that an inverse trigonometric function “undoes” what the original trigonometric function “does,” as is the case with any other function and its inverse. In other words, the domain of the inverse function is the range of the original function and vice versa.

For example, if $f(x)=\sin x$, then we would write $f^{-1}(x)=\sin ^{-1} x$. Be aware that $\sin ^{-1} x$ does not mean $\frac{1}{ \sin x}$. The following examples illustrate the inverse trigonometric functions:
1. $\sin (\pi / 6)=1 / 2$, then $\pi / 6=\sin ^{-1}(1 / 2)$
2. $\cos (\pi)=-1$, then $\pi=\cos ^{-1}(-1)$
3. $\tan (\pi / 4)=1$, then $(\pi / 4)=\tan ^{-1}(1)$

Domain & Range of Inverse Trigonometric Functions

The domain and the range of the inverse trigonometric function are added in the table below.

Differentiation of Trigonometric Functions

The derivatives of trigonometric functions gives the slope of the tangent of the curve. The derivative of $\sin x$ is $\cos x$ and here on applying the $x$ value in degrees for$\cos x$ we can obtain the slope of the tangent of the curve of $\sin x$ at a particular point. The formulas of differentiation of trigonometric functions are useful to find the equation of a tangent, normal, to find the errors in calculations.

$\frac{d}{d x}(\sin (\mathbf{x}))=\cos (\mathbf{x})$

$\frac{d}{d x}(\cos (\mathbf{x}))=-\sin (\mathbf{x})$

$\frac{d}{d x}(\tan (\mathbf{x}))=\sec ^2(\mathbf{x})$

$\frac{d}{d x}(\cot (\mathbf{x}))=-\csc ^2(\mathbf{x})$

$\frac{d}{d x}(\sec (\mathbf{x}))=\sec (\mathbf{x}) \tan (\mathbf{x})$

$\frac{d}{d x}(\csc (\mathbf{x}))=-\csc (\mathbf{x}) \cot (\mathbf{x})$

Integration of Trigonometric Function

The integration of trigonometric functions is helpful to find the area under the graph of the trigonometric function. Generally, the area under the graph of the trigonometric function can be calculated with reference to any of the axis lines and within a defined limit value. The integration of trigonometric functions is helpful to generally find the area of irregularly shaped plane surfaces.

$\int \cos x d x=\sin x+C$
$\int \sin x d x=-\cos x+C$
$\int \sec ^2 x d x=\tan x+C$
$\int \operatorname{cosec}^2 x d x=-\cot x+C$
$\int \sec x \cdot \tan x d x=\sec x+C$
$\int \operatorname{cosec} x \cdot \cot x d x=-\operatorname{cosec} x+C$
$\int \tan x d x=\log |\sec x|+C$
$\int \cot x . d x=\log |\sin x|+C$
$\int \sec x d x=\log |\sec x+\tan x|+C$
$\int \operatorname{cosec} x \cdot d x=\log |\operatorname{cosec} x-\cot x|+C$

Also Read,

Importance of Trigonometric Functions Class 11

Trigonometric Functions have a significant weighting in the IIT JEE test, which is a national level exam for 12th grade students that aids in admission to the country's top engineering universities. It is one of the most difficult exams in the country, and it has a significant impact on students' futures. Several students begin studying as early as Class 11 in order to pass this test. When it comes to math, the significance of these chapters cannot be overstated due to their great weightage. You may begin and continue your studies with the standard books and these revision notes, which will ensure that you do not miss any crucial ideas and can be used to revise before any test or actual examination.

How to Study Trigonometric Functions?

Start preparing by understanding and practising to find the value of the trigonometric functions. Try to be clear on concepts of trigonometric identities, differentiation and integration of trigonometric functions. Practice many problems from each topic for better understanding.

If you are preparing for competitive exams then solve as many problems as you can. Do not jump on the solution right away. Remember if your basics are clear you should be able to solve any question on this topic.

Important Books for Trigonometric Functions

Start from NCERT Books, the illustration is simple and lucid. You should be able to understand most of the things. Solve all problems (including miscellaneous problem) of NCERT. If you do this, your basic level of preparation will be completed.

Then you can refer to the book Trigonometry by SL Loney. Trigonometric Functions are explained very well in these books and there are an ample amount of questions with crystal clear concepts. Choice of reference book depends on person to person, find the book that best suits you the best, depending on how well you are clear with the concepts and the difficulty of the questions you require.