Sine and Cosine Graphs

This chapter explores sine and cosine graphs. It covers sketching sine and cosine graphs, finding angles which have s...

This chapter explores sine and cosine graphs. It covers sketching sine and cosine graphs, finding angles which have same sine or cosine values, and solving simple equations such as sin x = 0.6 You must have prior knowledge of basic trigonometry and sketching graphs attempting this chapter

Unit Circle – Sines

This section explains what exactly is the sine of an angle. First draw a circle of radius 1 as shown below. sine and cosine graphs-01 Then draw a radius at any angle that you wich as shown below. sine and cosine graphs-02 The sine of the angle is simply the height of the end of the line. In the above case we can see that when; sine and cosine graphs-42 …then… sine and cosine graphs-43 This is also true when you combine the circle idea above with the actual sine graph as shown below.
sine and cosine graphs-03
Above we can see that the angle is 48° or -312° that forms the height of the line to be 0.743. We say that; sine and cosine graphs-04 Below are notes to remember… The value of sinx is positive or negative as shown below. sine and cosine graphs-05 Remember the sine graph repeats every 360°

Cosine

The cosine curve works very similarly to what we have explored above except we look at the width of the curve and not the height. We best see the effect of the curve by rotating the unit circle above as shown below. sine and cosine graphs-06 We can compare it with the cosine graph to observe the effect.
sine and cosine graphs-07
Above we can see that the blue line is 61° or 299° while the width of the line is 0.485, We say that; sine and cosine graphs-08 Below are some notes to remember; The value of cosx is positive or negative as shown below. sine and cosine graphs-09 Also remember that the cosine graph repeats every 360°

Angles with the same sine value

You must also acknowledge that different angles may share the same sine value as you may have realised above. For example suppose we wanted to find the angle with the same sine as 50°. Below is the part of the sine curve with the 50° angle indicated. sine and cosine graphs-10 The sine curve is symmetrical about 90°. We can zoom in closer to get a clear observation as shown below. sine and cosine graphs-11 Above we have managed to find the sine as being 0.77. Below the blue line has been continued to show the other angle with the same sine. sine and cosine graphs-12 To find the other angle we simply take away 50° from 180°. sine and cosine graphs-13 So we can make a conclusion that; sine and cosine graphs-14 We can generalise by saying; sine and cosine graphs-15 Remember the graph is also continuos in the negative direction with the same pattern. Below the negative side of the graph has been shown.
sine and cosine graphs-16
We can find the other 2 angles which have the same sine as 50°. Below the blue line has been continued in the opposite direction.
sine and cosine graphs-17
Notice the gap above which is also 50° as we saw before. The first negative x value/angle is; sine and cosine graphs-18 The other gap is also 50° as shown below.
sine and cosine graphs-19
The final x value or angle is; sine and cosine graphs-20 We can conclude that all the following angles/values have the same sine; sine and cosine graphs-21 Notice above that the angles have been listed in ascending order. You can check then on your calculator to see whether this is true.

Angles with same cos value

Here we shall explore the cosine curve. Suppose we know that cos 45° = 0.707. What other angle gives the same value. The angle and sine have been indicated on the cosine graph below. sine and cosine graphs-22 We can continue the blue line to find the other angle. Remember that the graph continues in the same pattern in the negative x values.
sine and cosine graphs-23
All the following angles/values have the same cosine value; sine and cosine graphs-24 Note that the angles are written in ascending order.

Solving equations in form sinx = 0.64

Solving simply means write down all of the angles which have a sine 0.64. Suppose we had to solve; sine and cosine graphs-25 …in the range -360 ≤ x ≤ 360 Using the graph below we get 40° for sin-1 (0.64). You can use a calculator to get a very accurate value. sine and cosine graphs-26 If we continue the blue line we can see the next angle or we could simply use the method we discovered above; sine and cosine graphs-44 There are also other negative angles as we discorvered earlier. sine and cosine graphs-45 The final value of x or angle is; sine and cosine graphs-28 We have managed to solve the problem. If sinx = 0.64; sine and cosine graphs-29 We can see these angles on the graph below.
sine and cosine graphs-30

Example

Here is another problem that we could solve; sine and cosine graphs-31 …in the range -360 ≤ x ≤ 360e Remember solving simply means write down all the angles which have the sine of -0.57/ We can use the calculator to find the first angle; sine and cosine graphs-32 The first angle has been shown on the graph below. sine and cosine graphs-33 The other negative angle is 34.8° more than -180°, we write; sine and cosine graphs-34 Remember there is also two more positive angles or values. The first one is; sine and cosine graphs-36 The final one at; sine and cosine graphs-35 We have managed to solve the problem. We can now conclude that if sinx = -0.7. sine and cosine graphs-36 Above the angles have been rounded to the nearest degree. The angles have been shown on the graph below.
sine and cosine graphs-37

Solving equations such as cos x = 0.7

Now we shall look at cosine. Remember that solving simply means find all angles which have a cosine of 0.7. We can use the calculator to find the first solution or value.
sine and cosine graphs-38
Or we could use the graph shown below to find that it is the same;
sine and cosine graphs-39
There is also another positive angle besides 45.6° as we already know that; sine and cosine graphs-40 Remember that the cosine graph is symmetrical about the y-axis. So the other angles are simply -314° and -56° We have managed to solve the problem. We now know that if cos x = 0.7 then; sine and cosine graphs-41
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