How To Draw A Sine Graph

Graphing the Sine and Cosine Functions

Learning Objective(due south)

· Determine the coordinates of points on the unit of measurement circumvolve.

· Graph the sine function.

· Graph the cosine part.

· Compare the graphs of the sine and cosine functions.

Introduction

You know how to graph many types of functions. Graphs are useful because they can accept complicated information and display it in a elementary, easy-to-read manner. Y'all'll at present acquire how to graph the sine and cosine functions, and see that the graphs of the sine function and cosine office are very similar.

Values of the Sine and Cosine Functions

We take seen a point (10,y) on a graph of a office. The first coordinate is the input or value of the variable, and the second coordinate is the output or value of the part.

Each bespeak on the graph of the sine function will have the form , and each bespeak on the graph of the cosine part will have the class . It is customary to utilize the Greek letter of the alphabet theta, , as the symbol for the angle. Graphing points in the form is just like graphing points in the grade (ten, y). Along the ten-axis we volition exist plotting , and along the in the y-axis nosotros volition exist plotting the value of . The graphs that we'll draw will employ values of in radians. Before drawing the graphs, it'll be useful to find some values of and , and and then gather them together in a table.

Let's review the general definitions of these functions. Given any angle , describe information technology in standard position together with a unit circumvolve. The terminal side will intersect the circle at some bespeak , as shown below.

The value of has been defined to exist the x-coordinate of this point, and the value of has been defined to exist the y-coordinate of this point.

Case
Problem	Detect the values of and for .
		Yous might find it useful to convert these angles to degrees. All iv angles have a reference angle of 30° or radians.
		Use the right triangle definition to observe the and for .
		Graph the 4 angles in standard position. The coordinates of the point in the first quadrant were found above. The x-coordinate is the value of cos θ, and the y-coordinate is the value of sin θ. The other points are reflections of the offset point over the x-centrality, the y-axis, or both.
Reply	, , ,

You can go through a similar process to observe the values of and for . All four angles take a reference angle of radians or 45°.

Using the fact that gives y'all the coordinates of the point in the first quadrant. Since the other points are reflections of this one, the coordinates have the same or the reverse values.

The diagram below tin be used to discover the values of and for . Note that because , when yous draw the angle in standard position, y'all finish up dorsum at the ten-axis. radians or corresponds to the aforementioned indicate as 0 radians does, namely .

Using the coordinates of the four points gives you lot:

To get more familiar with the coordinates of points on the unit circle, try the following interactive practice:

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please get to www.java.com

The Graph of the Sine Function

Our goal correct at present is to graph the function . Each bespeak on this office's graph will have the grade with the values of in radians. The start step is to gather together in a table all the values of that you know. To get-go nosotros will use values of θ between 0° and 180° .

(in degrees)	(in radians)
0°	0	0
30°
45°
60°
90°		1
120°
135°
150°
180°		0

When we graph functions, nosotros often say to graph the function on an interval. We use interval notation to describe the interval. Interval annotation has the form , which means the interval begins at a and ends at b. In the example, the notation has the same meaning as .

Example
Trouble	Graph the sine function on the interval . Depict the values of the function every bit goes from 0 to .
		Plot all the points from the concluding cavalcade of the table above. Annotation that and that . Connect the points with a smooth curve.
Answer	The values increment from 0 to 1 and then decrease from ane to 0.

Note that our input is , the measure out of the angle in radians, and that the horizontal centrality is labeled , non 10. Next we will gather together all the values of that yous know for in one table.

(in degrees)	(in radians)
180°		0
210°
225°
240°
270°
300°
315°
330°
360°		0

You could simply plot all the points from the concluding column and keep the graph in the last case. But notice the following: the values in the 3rd cavalcade (or y-coordinates of the points) have the reverse values of the points that nosotros simply graphed. This ways that instead of plotting points in a higher place the -axis, y'all will be plotting points below the -centrality. Also, the inputs and outputs are spaced out in the same fashion for this function of the graph as they were for the first office of the graph. So instead of having a "hill" that goes from 0 up to 1 and down to 0, you will take a "valley" that goes downwards from 0 to and and then up to 0.

We have used values of θ from 0 to 2 π to draw the graph of the sine office. What does the graph look similar if we continue to the correct of two π for , which is i more time around the unit of measurement circle? In terms of degrees, these are angles between 360° and 720°. Permit's go dorsum and look at 1 of these angles in standard position with the unit circle. A 400° angle is shown below.

Because , the angle travels a full rotation plus some other 40°, every bit shown by the curved arrow in the diagram. Imagine y'all are on (1, 0) and you walk effectually the circle one complete time and and then you walk a petty more to end up at (x, y). This point where the terminal side intersects the unit of measurement circumvolve is the same point that you would become for a twoscore° angle. All of the trigonometric functions for these 2 angles are computed using the coordinates of this betoken. This means that , , , and the same is true for the three reciprocal functions.

There is nothing special about 400°. Yous could draw other angles that are greater than 360° and find similar results. The results in a higher place for sine and cosine can be rewritten as and . In general, it is truthful that and , or, using radians:

These 2 equations tell usa that when we go effectually the circle a second fourth dimension, we are going to get the same values for as nosotros did for and the same values for as nosotros did for . In other words, as nosotros get around the aforementioned circle a 2nd time, at the aforementioned locations on the circle nosotros will get the same values for the y-coordinate and the 10-coordinate that we did the first time around the circle.

Instance
Problem	Sketch the graph of the sine office on the interval and find the range.
	Because the values of the sine function between and are the same equally the values betwixt 0 and , the shape of the graph between and is the same as the shape of the graph between 0 and . The range is the set of all y-values that the function can have, so the range of is .
Answer	The range of is .

The same reasoning that we used above works for negative angles. For example, the angles and 135° are drawn in standard position with the unit circle below.

U19_L2_T2_8

Because they are coterminal angles, they intersect the unit of measurement circle at the same signal and therefore have the same coordinates. Therefore, , , and so on for the other trigonometric functions. Notice that we could rewrite the starting time equation equally . Too, , or , is true for any bending including negative angles. The equation tells u.s. that each fourth dimension we go one boosted full revolution around the circumvolve we go the aforementioned values for the sine and the cosine as we did the first time around the circle.

Example
Problem	Sketch the graph of the sine function on the interval .
	Because is true for negative angles also as positive angles, the values of the sine function betwixt and 0 are the same every bit the values of the sine part between 0 and . Therefore, the shape of the graph between and 0 is the same as the shape of the graph between 0 and .
Respond

Since the equation is true for any bending, it is an identity. Nosotros can use this identity to continue the graph of the sine part in either direction. This "loma and valley" pattern over an interval of length will continue in both directions forever.

Each time we add 2 π to an bending, say , we will become the same part value.

You can also utilise the identity to a higher place to simplify calculations of the sine function by repeatedly subtracting from the angle. For example:

What is the value of ?

Show/Hide Reply

Correct. Rewrite as . The represents having gone effectually the circumvolve twice. Using the identity to remove i revolution at a time, nosotros become .

Incorrect. Maybe y'all simplified incorrectly and thought this was equal to . Utilize the identity to simplify . The right answer is .

Incorrect. You may have incorrectly converted from degrees to radians or confused sine and cosine. Use the identity to simplify . Recollect that . The correct answer is .

Wrong. You may accept simplified incorrectly and thought this was equal to . Employ the identity to simplify . The right reply is .

The Graph of the Cosine Function

At present our goal is to graph . We will get through the same procedure as we did for the sine function, and the result volition be like.

Each point on the function's graph will have the form with the values of in radians. The first step is to gather together in a tabular array all the values of that you know. To start nosotros will utilise values of θ between 0° and 180° .

(in degrees)	(in radians)
0°	0	1
30°
45°
60°
ninety°		0
120°
135°
150°
180°

Case
Trouble	Graph the cosine function on the interval . Describe the values of the role as goes from 0 to .
		Plot all the points from the terminal column of the table above. Note that and that . Connect them with a smooth curve.
Answer	The values decrease from 1 to 0 and as well continue to decrease from 0 to .

One time once again, our input is , the measure of the angle in radians, and the horizontal centrality is labeled , not x. Adjacent nosotros will gather together all the values of that you know for in 1 table.

(in degrees)	(in radians)
180°
210°
225°
240°
270°		0
300°
315°
330°
360°		ane

Again, you could only plot all the points from the last column and continue the graph. Instead, compare the values in the 3rd columns of the ii tables: they are the aforementioned numbers, but in the reverse order. These are the y-coordinates of the points. This means that while the first function that we graphed decreased from 1 downwards to , this 2nd part that we are graphing will increase from to i and accept the "aforementioned shape" (turned around the other way). Here it is:

The side by side step is to keep the graph for input values . When we were in the process of graphing the sine office, we established the following identity:

This equation tells us that when we become around the circle a second fourth dimension, nosotros are going to get the same values for as we did for .

In other words, as we go around the same circumvolve a second time, at the same locations on the circle we will get the same values for the 10-coordinate that we did the get-go fourth dimension around the circumvolve.

Instance
Problem	Sketch the graph of the cosine part on the interval .
	Considering the outputs between and are the same as the outputs between 0 and , the shape of the graph between and is the same every bit the shape of the graph between 0 and .
Reply

Merely as the identity is true for negative angles, the identity is also true for whatever negative bending .

Example
Problem	Sketch the graph of the cosine role on the interval .
	Because is true for negative angles as well as positive angles, the values of the cosine office between and 0 are the same as the values of the cosine function between 0 and . Therefore, the shape of the graph between and 0 is the same as the shape of the graph between 0 and .
Answer

The identity was used above to extend the graph of the cosine function to the correct and to the left. You tin can apply this to continue to extend the graph in both directions. You will go another "loma and valley" pattern that repeats later intervals of length in both directions.

Another of import feature of the graph of is that the left and right halves are mirror images of each other over the y-axis. The graph of has this aforementioned property. Another mode to describe this is to say that if you substitute a number and its opposite into the office, you volition get the same value every bit the previous equation. For example, , , or in general, . We say that the graph is symmetric virtually the y-centrality. The diagram below shows two points taken from a symmetric graph.

U19_L2_T2_14

The summit of the points at opposite inputs is the same. The height is the value of the function. A function whose graph is symmetric about the y-axis has .

What is the range of the cosine function?

A) all values in the interval

B) all values in the interval

C) all values in the interval

D) all real numbers

Show/Hide Answer

A) all values in the interval

Wrong. You were probably looking at the y-values, which is the right thing to do. However, you chose only role of the range. The range is the set of all y-values that the part can accept; in this case that would be . The correct answer is B.

B) all values in the interval

Correct. The graph of the role extends forever in both directions, so its domain is all existent numbers. It consists of a repeating hill and valley pattern. The valley goes down to a y-value of , and the colina goes upward to a y-value of 1. All y-values between these two y-values are also outputs of the function. And so the set of outputs, or range, is all numbers from to ane.

C) all values in the interval

Incorrect. This interval, as a prepare of inputs, will give you one complete design. Information technology is the set of outputs that you are looking for. The right answer is B.

D) all real numbers

Incorrect. Perhaps you were thinking of the domain of the function, which is all existent numbers. The range is the gear up of all outputs or y-values. The right answer is B.

A Comparison of the Graphs of Sine and Cosine

The graphs of sine and cosine both take hills and valleys in a repeating pattern.

Since this repeating pattern tin can exist extended indefinitely to the left and right, the domain for both functions is the existent numbers. The range for both of them is the interval .

Let'due south now compare these graphs in some other ways.

First we want to see what happens to the graph of a function when you modify the input past adding a constant to it. Compare and . Hither is a tabular array with some values for these two functions.

(in radians)		(in radians)
0	0		1



	i		0



	0

Now we'll graph these 2 functions. As a reminder, the input is for both of these functions. To graph , you use the numbers in the first and 2nd columns. To graph , you use the numbers in the first and fourth columns. (The third column was but written as a convenient intermediate footstep. Y'all do non use that for the graph.)

Outset observe, as shown past the pieces in red, that the effect of adding to the input is to shift the graph to the left by units. Y'all may think seeing this effect when you graphed radical functions such as and (adding the 1 to the input shifted the graph of to the left 1 unit). In general, if you lot add a positive abiding c to the input of a role, that volition have the issue of shifting the original graph to the left by c units. If y'all decrease a positive constant c from the input of a function, that will have the result of shifting the original graph to the right past c units.

Next observe that the graph on the right is already familiar to you. It is the graph of ! So you can say that the graph of is the same as the graph of , or you tin say that the graph of shifted to the left by units is the graph of .

The next instance looks at a shift in the other direction.

Example
Problem	Sketch the graph of on the interval . How does the graph compare to the graph of ?
	The graph of is the same as the graph of shifted units to the correct.

Answer	The graph of is the aforementioned every bit the graph of .

Because the patterns echo, yous could start with the graph of either sine or cosine and shift it past unlike amounts to the right or left to become the graph of the other function.

Which comparing of the graphs of and is truthful?

A) They are the same.

B) The graph of shifted units to the correct is the graph of .

C) The graph of shifted units to the correct is the graph of .

D) The graph of shifted units to the left is the graph of .

Show/Hide Respond

A) They are the same.

Incorrect. The two graphs have the same repeating pattern or the same general shape, but they are not identical. The right reply is D.

B) The graph of shifted units to the correct is the graph of .

Wrong. If you switch sine and cosine in this selection you get a correct statement. The ii graphs described in this choice have the same shape, but the hills and valleys do not friction match up. The correct respond is D.

C) The graph of shifted units to the right is the graph of .

Wrong. If you shift the graph of the sine function by units to the right, you become a graph that "begins" (at ) with a valley. This is non the graph of the cosine function. The right answer is D.

D) The graph of shifted units to the left is the graph of .

Correct. If you shift the graph of by units to the left, you volition become a graph that "begins" (at ) at the peak of a hill. This is the graph of .

For practice graphing sine and cosine, endeavour the following interactive exercise:

This is a Coffee Applet created using GeoGebra from www.geogebra.org - information technology looks like you don't take Java installed, please go to world wide web.java.com

Summary

The graphs of sine and cosine have the aforementioned shape: a repeating "loma and valley" pattern over an interval on the horizontal axis that has a length of . The sine and cosine functions have the same domain—the real numbers—and the same range—the interval of values .

The graphs of the 2 functions, though similar, are not identical. 1 way to describe their human relationship is to say that the graph of is identical to the graph of shifted units to the left. Another way to depict their relationship is to say that the graph of is identical to the graph of shifted units to the right.