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Changes in Sine and Cosine Functions

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Sine and cosine Functions are Circular Functions or we can say they are trigonometric functions. Here we will discuss Changes in Sine and Cosine Functions. Sine function is a trigonometric Ratio which is equals to ratio of perpendicular and hypotenuse of a right angled triangle and cosine function is equals to ratio of base and hypotenuse of a right angled triangle.
Relation between sine and cosine function can be stated with help of Pythagorean trigonometric identity. According to Pythagorean trigonometric identity: Sin 2 x + Cos 2 x = 1.
Using this Pythagorean identity we can change sine function into cosine function and cosine function into sine function. This conversion can be done as shown below:
Sin 2 x + Cos 2 x = 1,
Sin 2 x = 1 – cos 2 x,
Sin x = √ (1 – cos 2 x),

Similarly,
Cos 2 x = 1 – sin 2 x,
Cos x = √ (1 – sin 2 x),

This is how conversion of sine and cosine functions can be done using Pythagorean trigonometric identity.

Period

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Period of any trigonometric function defines the angle at which a value gets repeats. Suppose a parameter has value one at angle 35 and if we increase the value of the angle and at 135 angle we again get the value as one. After 135 all the values repeats again and again then we can say that the function has a period of 100 as 135-35 = 100. In other words, we can say that, at a fix interval a function repeats its value this is called as period of the function.
Now, if we talk about period of sin function, then the period of sign function shows the cycle length of the curve and height, which is a very essential characteristic of the waveform. The period of sign function is 2π; here the meaning of π is 180 degree. We can also determine the period of any function by its graph just we need to check in which interval the shape of graph get repeats.
If we talk about sine function then we know that, if we have sine function at an angle zero and has value 0, then it has the zero value on 2π, and if we calculate the angle 0 + 30 then it will be equal to 2π + 30. The value of sin 30 is 1/2 and value of sin 390 will also be 1 /2. All the values will be same and again after 2π angle the value will be zero.
If we talk about period of cosine then at angle zero it is has value as 1 and then on angle 2π it is again having the value 1, so its period will again 2 π, so we can say that period of sine and cosine are same that is 2 π.
In this way, we can find the period of any trigonometric function we just need to check where the value gets repeated.

Steps For Sine & Cosine With Translations

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Sin and cos Functions are short form of sine and cosine respectively and are most important trigonometric functions. All other trigonometric functions (like tan, cot, sec, cosec etc.) can be expressed in form of these two trigonometric functions.
Sin and cos functions are defined by Right Triangle as shown below in diagram:

Where 'x' is the angle.
Here sine and cosine are defined mathematically as:
Sin x = (perpend / hypotenuse),
Cos x = (base / hypotenuse),


Translations Of Sine And Cosine Functions can be of two types:
1) Horizontal Translations
2) Vertical Translations
1) Horizontal translations
b = f (a - t) is called the horizontal translation. This can also be called a phase shift.
If written (a - t) then it indicates that shift in 't' units to right and if (a + t) then it shows that shifting is 't' units to left where 't' is |t|.


2) Vertical Translations
Lets consider following equation to understand translations
b = c + f (a) shows vertical translations of b = f(a) up or down. If ‘c > 0’, shifts 'c' units up and if ‘c <0’, shifts 'c' units down.

Steps for Translation Of Sine And Cosine Functions:
1. First of all search that interval whose length is one Period by solving three part inequality.
2. Then interval must be Divided into four equal parts.
3. Generate the function for all five x- values which are obtained from step 2.
4. Plot all five points on graph and connect all of them with suitable curve.
5. Sketch the graph over additional periods, as required.

Another method is to Graphing b = sin ta or b = c cos ta. Amplitude of above function is |c| and period is (2π / t). Now by using Translation Of Sine And Cosine Functions, graph of original function can be plotted.