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Trigonometric Functions

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Trigonometric functions are the functions of angles. They are also known as circular functions. Trigonometric functions are helpful to form a relation between the angle and sides of a right angled triangle. Most familiar trigonometric function are the sine, cosine and tangent. The trigonometric functions can be more precisely defined with the help of ratios, ratios of the two sides of a right angle triangle.

These functions are also used for navigation, engineering, optics, probability theory.

The functions such as sine and cosine are apparently used to model periodic function phenomenon like in sound and light waves. Basically, there are six common trigonometric functions. We can relate them easily with one another. They can be defined as the ratios of two sides of a right triangle containing the angle.

Six Trigonometric Functions

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Let us define the trigonometric functions with the help of angle ‘$\theta$’, initiating with the right angled triangle. The three sides of triangle can be named as follows:

Six Trigonometric Functions


Hypotenuse: The longest side of triangle and it is the side opposite to the right triangle.
The opposite side is the side opposite to the angle we are interested in (angle A).
The adjacent side is the side having both the angles of interest.

The total of angle inside the right triangle is 180$^{\circ}$. Therefore, the angles of right triangle will be 90$^{\circ}$ and the other two angles will be acute whose sum will be equal to 90$^{\circ}$. So, the following description of angles will apply to the angles between the 0$^{\circ}$ - 90$^{\circ}$.
  • Sin $\theta$ = $\frac{\text{Opposite}}{\text{Hypotenuse}}$
  • Cos $\theta$ = $\frac{\text{Adjacent}}{\text{Hypotenuse}}$
  • Tan $\theta$ = $\frac{\text{Opposite}}{\text{Adjacent}}$
  • Csc $\theta$ = $\frac{\text{Hypotenuse}}{\text{Opposite}}$
  • Sec $\theta$ = $\frac{\text{Hypotenuse}}{\text{Adjacent}}$
  • Cot $\theta$ = $\frac{\text{Adjacent}}{\text{Opposite}}$
Inverse trigonometric functions are the inverse functions of the trigonometric functions with restricted domains because for any given input, there can exist more than one output. That is, for a given number, there exists more than one angle whose sine, cosine, etc., is that number and the symbol for the inverse functions differ from the symbols for the inverse relations.

The chart below shows the restricted ranges that transform the inverse relations into the inverse functions:

Inverse Trigonometric Functions
Domain
Range
y = arcsin(x) -1$\leq$ x $\leq$ 1 -$\frac{\pi}{2}\leq y\leq \frac{\pi}{2}$
y = arccos(x) -1$\leq$ x $\leq$ 1 0$\leq y\leqslant \pi$
y = arctan(x) Real Numbers -$\frac{\pi}{2}< y< \frac{\pi}{2}$
y = arccsc(x) 1 $\leq$ x or x $\leq$ -1 -$\frac{\pi}{2}\leq y\leq \frac{\pi}{2}$, y $\neq$ 0
y = arcsec(x) 1 $\leq$ x or x $\leq$ -1 0 $\leq$ y $\leqslant \pi$, y $\neq\frac{\pi}{2}$
y = arccot(x) Real Numbers 0 < y < $\pi$

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Properties of Trigonometric Functions

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Given below are some of the properties of trigonometric fuctions:

1. Sin and cosine are periodic functions of period 360$^{\circ}$. It does not change the angle.
Sin (x + 360$^{\circ}$) = Sin (x)
Cos (x + 360$^{\circ}$) = Cos (x)

2. The Pythagorean Theorem: The square of the hypotenuse is the sum of the squares of the legs. The other two sides are referred to as the legs of the triangle.

3. Half angles: Used to find the values of unknown trigonometry functions.
Sin($\frac{x}{2}$) = $\sqrt{\frac{1-\cos x}{2}}$
Cos($\frac{x}{2}$) = $\sqrt{\frac{1+\cos x}{2}}$
Tan($\frac{x}{2}$) = $\sqrt{\frac{1-\cos x }{1 + \cos x}}$

4. Sine and cosine are complementary.

5. Sine and cosine are odd and even functions respectively.
Odd Function: A function f is said to be an odd function, if for any number x, f(-x) = -f(x).
Even Function: A function f is said to be an even function, if for any number x, f(-x) = f(x).

Periodic Functions

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A function that repeats its values in regular intervals or periods is known as periodic function. They are used to describe oscillations, waves, and other phenomenon that exhibit periodicity.

A function f(x) is said to be periodic with period p if f(x) = f(x + np), n = 1, 2, 3..... A function with period P will repeat on intervals of length P, and these intervals are referred to as periods.

The following table summarizes the names given to periodic functions based on the number of independent periods they posses:

Number of Periods Nature of Functions
1 Single periodic function
2 Doubly Periodic function
3 Triply Periodic function

Trigonometric Functions of Any Angle

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Trigonometric functions are used to calculate the internal angles of triangles. Trigonometric functions are sine function, cosine function, tangent function, cotangent function, cosecant function and secant function. These functions give a definite value of angles in certain degrees. We can find the trigonometric functions of any angle.

Let us go through certain examples in which we will calculate trigonometric functions of some specific angles. Let us take first angle as 00.

Sin 0$^{\circ}$ = 0
Cos 0$^{\circ}$ = 1
Tan 0$^{\circ}$ = 0
Cot 0 $^{\circ}$ = not defined
Sec 0$^{\circ}$ = 1
Csc 0$^{\circ}$ = not defined

So, we can see that values of cot and csc functions are not defined.

Similarly, let us calculate values of all above six functions for angle 30$^{\circ}$. Here, we will find each of the values as below:

Sin 30$^{\circ}$ = $\frac{1}{2}$
Cos 30$^{\circ}$ = $\frac{\sqrt3}{2}$
Tan 30$^{\circ}$ = $\frac{1}{\sqrt3}$
Cot 30$^{\circ}$ = $\sqrt{3}$
Sec 30$^{\circ}$ = $\frac{2}{\sqrt3}$
Cosec 30$^{\circ}$ = 2

So, all these values are predefined as per trigonometric function standards.
Similarly, we can calculate values for angles in radians also. We generally denote value of a radian as '$\pi$ '. In degrees, value of '$\pi$' is 180$^{\circ}$.

For all trigonometric functions, we calculate value of $\frac{\pi}{2}$ as shown below:

Sin $\frac{\pi}{2}$ = 1
Cos $\frac{\pi}{2}$ = 0
Tan $\frac{\pi}{2}$ = not defined
Cot $\frac{\pi}{2}$ = 0
Sec $\frac{\pi}{2}$ = not defined
Cosec $\frac{\pi}{2}$ = 1

Tan and Sec values of $\frac{\pi}{2}$ are not defined as per standard predefined rules of trigonometry.

Trigonometric Functions Table

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The table of trigonometric angles is shown below.

X

0$^{\circ}$

30$^{\circ}$

45$^{\circ}$

60$^{\circ}$

90$^{\circ}$

180$^{\circ}$

270$^{\circ}$

Sin x

0

$\frac{1}{2}$

$\frac{1}{\sqrt{2}}$

$\frac{\sqrt{3}}{2}$

1

0

-1

Cos x

1

$\frac{\sqrt{3}}{2}$

$\frac{1}{\sqrt{2}}$

$\frac{1}{2}$

0

-1

0

Tan x

0

$\frac{1}{\sqrt{3}}$

1

$\sqrt{3}$

Not defined

0

Not defined

Cot x

Not defined

$\sqrt{3}$

1

$\frac{1}{\sqrt{3}}$

0

Not defined

0

Sec x

1

$\frac{2}{\sqrt{3}}$

$\sqrt{2}$

2

Not defined

-1

Not defined

Csc x

Not defined

2

$\sqrt{2}$

$\frac{2}{\sqrt{3}}$

1

Not defined

-1

Graphing Trigonometric Functions

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Trigonometric functions can be defined as the functions of angles. They are also known as circular functions. Basically, they are used to compare angles to length of sides of triangle. Here, we will see graphing trigonometric functions.

We will start with basic sine function. Suppose, we have a function f(x) = sin (x). Amplitude value of this function is 1. This is because, graph of sine function always goes one unit up and one unit down from given midline of graph. Time period defined for sine function graph is 2$\pi$. This is because, wave of sine function changes after every 2$\pi$ units. Graph of trigonometric sine function is shown below:

Graphing Trigonometric Functions
Graphing Trigonometric Function
This graph is three times taller than the above graph. In this graph, amplitude value changes from 1 to 3. Amplitude value of the above function is defined as ‘1’. Whatever number is given for 'x', it is multiplied with trigonometric function which results in amplitude value. In this case, number is taken as 3 (because amplitude value is 3). So, 0.5 cos (x) will have amplitude of $\frac{1}{2}$, and -2 cos (x), it has amplitude of 2 and also be flipped upside down.

In mathematics, commonly used trigonometric functions are sine, cosine and tangent function.

Derivatives of Trigonometric Functions

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Trigonometric functions derivatives is basically differentiation of trigonometric functions to find the rate at which the function changes with respect to a variable. Six trigonometric functions that we are going to differentiate are Sin (x), Cos (x), Tan (x), Csc (x), Sec (x) and Cot (x). One should have good knowledge of various differentiation rules to find derivatives of trigonometric functions.

Basic derivatives are considered to be of sin (x) and cos(x). If these are known to one, others can easily be evaluated using trigonometric formulae because rest can be represented in terms of sine and cosine.

Given below are some general formulae which can be used in complex situations involving trigonometric functions.
  • Sin'(x) = Cos(x)
  • Cos'(x) = -Sin (x)
  • Tan'(x) = Sec$^{2}$ (x)
  • Cot'(x) = -Csc$^{2}$ (x)
  • Sec'(x) = Sec(x)Tan(x)
  • Csc'(x) = - Csc(x)Cot(x)

Integral of Trigonometric Functions

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Suppose we have an odd function f that is periodic with period 2L and that is integrable on any interval. Let us say that the function g is defined as
g(x) = $\int_{0}^{x}$ f(t)dt

Integration of basic trigonometric functions are as follows:

$\int$ Cos x dx = Sinx + c
$\int$ Sin x dx = - Cos x + C
$\int$ Sec$^{2}$ x dx = Tan x + C
$\int$ Csc$^{2}$x dx = - Cot x + C
$\int$ Sec x Tan x dx = Sec x + C
$\int$ Csc x Cot x dx = - Csc x + C
If f is continuous and periodic with period a, then $\int_{0}^{a}$ f(t)dt = $\int_{b}^{b + a}$ f(t)dt for all b $\epsilon \mathbb{R}$

Trigonometric Functions Problems

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Given below are some of the example problems based on trigonometric functions.

Solved Examples

Question 1: Consider y = Sin 8x.
  1. What does the 8 indicate?
  2. Find the period of the given function.
  3. Where are its zeros?

Solution:
1. In an interval of length 2$\pi$, there are 8 periods.

2. Period = $\frac{2\pi}{8}$
= $\frac{\pi}{4}$

3. Zeros will be at x = $\frac{n\pi}{8}$

Question 2: Given Cos $\theta$ = $\frac{15}{19}$, find Sec $\theta$
Solution:
As sine and cosine are reciprocal functions, we have

Sec $\theta$ = $\frac{1}{\cos \theta}$

= $\frac{19}{15}$