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# Trigonometric Functions

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 Sub Topics Trigonometric Functions are also known as circular Functions; trigonometric functions are the functions of angles. These functions are helpful to form a relation between the angle and sides of triangle. Sine, cosine, tangent, are the function which are most familiar trigonometric function. The trig function can be more precisely defined with the help of ratios, ratios of the two sides of a Right Angle triangle. Trigonometric functions can be used for calculating lengths and angle of the right angle triangle. Trigonometric functions can be used for navigation, medical imaging, engineering, optics, probability theory. Also the function such as sine and cosine are apparently used to model periodic function phenomenon like in sound and light waves. Basically six common trigonometric functions are there. We can relate them easily with one another. Now lets define the trigonometric functions with the help of angle ‘C’, initiating with the right angled triangle that consists of angle ‘C’. The three sides of triangle can be named as: Hypotenuse is the longest side of triangle and it is the side opposite to the Right Triangle. The opposite is the side opposite to the angle we are placing in i.e. angle ‘C’. The total of angle inside the right triangle is 1800, therefore the angles of right triangle will be 900 and other two angles will be acute whose sum will be equal to 900. So the following description of angles will apply to the angles between the 00-900. Trig functions are discussed below: Sine C = Perpendicular/hypotenuse. Cosine C = base/ hypotenuse. Tangent C = perpendicular/ base. Cosecant C = 1/ sine C Secant C= 1/ Cos C Cotangent C = 1/ Tan C = cos C/ sin C.

## Periodic Functions

In Calculus, we use the notion of periodic Functions frequently. A periodic function can be defined as a function whose values reiterate in recurrent spaces or intervals. There are a large number of examples of the periodic Functions that include the trigonometric functions (i.e. sine, cosine, tangent, and cotangent and secant functions). A notable feature of the functions is that they reiterate over periods of 2π radians. Periodic functions are applied all over the science and Math to depict phenomenon that display periodicity such as waves.
To understand the concept of periodic functions, let us take a function f (a). If the function is a periodic function, then there will exist some value h such that f (a + h) = f (a). For example, let us see the sine function.
Sin (a + 2 π) = sin a.
This means that sin (0 + 2 π) = sin (0) = 0, or sin (π / 3 + 2 π) = sin (π / 3) = √3 / 2. To illustrate it further we can say, if the value of a function at Point a is some ‘b’ then for it to exhibit periodicity, there will be a quantity h such that the value of the function at point a + h will also be b. The value h is termed as Period of the function. Thus, the period of a function is a value after which it repeats its behavior.
Based on the number of periods, periodic functions can be classified into singly periodic functions, doubly periodic function, and triply periodic function and so on. The fundamental or prime period of a function is the extent of a least uninterrupted part of the Domain for one complete cycle of the function. In other words, it is the smallest period that can be possible for the given function. For the sine function, the fundamental period is 2π.

## Evaluation of Trigonometric Functions

Trigonometry is a stream of mathematics which performs operations on angles and sides of a triangle. Trigonometric Functions are defined as the Functions including the various Trigonometric Equations. There are six Trigonometric Functions defined in Trigonometry these are:
sin x, cos x, tan x, cot x, sec x and cosec x.
Cosec x is inversely proportional to sine x. If for any function the value of sin x= ½ then it will be reciprocated for cosec x and the value will be obtained as 2. The identities defined for evaluation of trigonometric functions are as follows:
sin2x + cos2 x=1
sec2x – tan2x =1
cosec2 x -cot2 x =1
Reciprocal identities are as follows:
sin x= 1/cosec x
cos x= 1/sec x
tan x= sin x / cos x
cot x= cos x / sin x
To evaluate trigonometric functions, one must know about the values of function at those different intervals or at different points. The value of trigonometric functions is different at different points.
Six trigonometric functions are evaluated using same above equations.
Let’s understand it using an example, suppose the value of sin x= 1/2 and we need to find the value of cot x. then using the identity cosec x= 1/sin x we get, cosec x= 2. Now using identity, cosec2x – cot2 x=1
cot2x=cosec2x-1 that gives, cot2x= 4-1=3 and cot x= √3
In the above example, we are given with the value of sin and for determining cot function; cosec is used as an intermediate function. The value of cot can be determined by cosec. Thus, using the identities various trigonometric functions can be determined.
Evaluation of trigonometric functions is not a difficult task to do; we just need to make proper use of identities at right place. Imaginary values cannot be defined for trigonometric functions as their values are not specified yet.

## Sin Cos Tan Table

The table of trigonometric angles is shown below.

 X 00 300 450 600 900 1800 2700 Sin x 0 ½ 1 / √2 √3 / 2 1 0 -1 Cos x 1 √3 / 2 1 / √2 ½ 0 -1 0 Tan x 0 1 / √3 1 √3 Not defined 0 Not defined Cot x Not defined √3 1 1 / √3 0 Not defined 0 Sec x 1 2 / √3 √2 2 Not defined -1 Not defined Cosec x Not defined 2 √2 2 / √3 1 Not defined -1

We can also write the value of sin 300 as / 6 and sin 450 as / 4, sin 600 = / 3,

cos 300 = / 6, cos 450 = / 4, cos 600 = / 3,

tan 300 = / 6, tan 450 = / 4,

Fundamental relation is also useful to solve the Trigonometric Functions directly:

 Fundamental relation sin p = 1 / cosec p, cos p = 1 / sec p, tan x = sin x / cos x, cot x = cos x / sin x sec x = 1 / cos x cosec x = 1 / sec x

## What are the Derivatives of Trigonometric Derivatives

Trigonometric Functions Derivatives is basically differentiation of Trigonometric Functions to find the rate at which function changes with respect to a variable. Six trigonometric functions that we are going to differentiate are sin (X), cos (X), tan (X), cosec (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. Different rules of differentiation related to constants, exponents, fractions etc. are important to imitate the calculation for derivatives of trigonometric functions. Here, are some general formulae which can be used in complex situations involving trigonometric functions.
D (sin X) / DX = cos X,
D (cos X) / DX = - sin X,
D (tan X) / DX = D (sin X / cos X) / DX,

Using rules for differentiating a fraction in above situation:
= (cos2 X + sin2 X) / cos2 X = 1 / cos2 X = sec2 X,
D (cot X) / DX = D (cos X / sin X) / DX = (- cos2 X - sin2 X) / sin2 X = - (1 + cot2 X) = - cosec2 X,
D (sec X) / DX = D (1 / cos X) / DX = (sinX / cos2 X) = (1 / cos X) * (sin X / cos X) = sec X tan X,
D (cosec X) / DX = D (1 / sin X) / DX = - (cosX / sin2 X) = - (1 / sin X) * (cos X / sin X) = - cosec X cot X.

## Trigonometric Functions of Any Angle

Trigonometric Functions are basically most important concept of Trigonometry which are generally used to calculate 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 00 = 0,
Cos 00 = 1,
Tan 00 = 0,
Cot 00 = not defined,
Sec 00 = 1,

Cosec 00 = not defined.
So we can see that values of cot and cosec functions are not defined. Now similarly let us calculate values of all above six functions for angle 300.
Here we will find each of the values as below:
Sin 300 = ½,
Cos 300 = 3/2,
Tan 300 =1/3,
Cot 300 = 3,
Sec 300 = 2/3,
Cosec 300 = 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 'π'. In degrees value of 'π' is 1800.
Let us calculate the values of π/2. As value of 'π' is 1800 so half of 'π' I.e. π / 2 is 180 / 2 that is 900. So for all trigonometric functions we calculate value of π/2 as shown below:
Sin π/2 = 1,
Cos π/2 = 0,
Tan π/2 = not defined,
Cot π/2 = 0,
Sec π/2 = not defined,
Cosec π/2 = 1.
We can see that tan and sec values of π/2 are not defined as per standard predefined rules of trigonometry.