Graph of sin X

We could create a table of various values of sin X as we increase the value of X from 0 to 360.

X	sin X
0	0
30	1
45	0.707
60	0.866
90	1
120	0.866
135	0.707
150	0.5
180	0
210	-0.5
225	-0.707
240	-0.866
270	-1
300	-0.866
315	-0.707
330	-0.5
360	0

We can plot these values on a graph and join up the curve between the points. The graph of Y = sin X is called a sin curve or a sin wave.

We can see that the value of sin X starts at 0 and goes between 1 and -1.

It is zero when X is 0, 180° and 360°

It is one when X is 90°

It is minus one when X is 270°

It is 0.5 when X is 30°, 150°, 210° and 330°.

Let’s add the other values from the table to the x and y axes. I’ve drawn dotted lines where sin X has positive and negative values of 0.5, 0.707, 0.866 and 1.

0.707 = √2 / 2

0.866 = √3 / 2

I’ll change the information in the graph below

The information can also be shown using a triangle in a circle

Remember that sin θ = O / H. sin θ is just the ratio of the side opposite angle θ to the hypotenuse in a right angled triangle. If the hypotenuse is 1 then sin θ = O.

Here is a unit circle. A unit circle is just a circle with a radius of 1.

Look at right angled triangle ABC within the unit circle.

Angle θ is 30°

The hypotenuse is 1

The side opposite angle θ is 0.5

So, sin 30 = 0.5

When θ is 45° the side opposite angle θ is 0.707. This is √2/2

When θ is 45° the adjacent side and the opposite side to θ are the same length. Also, angle BAC is equal to angle ABC. They are both 45°.

When θ is 60° the side opposite angle θ is 0.866. This is √3/2

When θ is 60° angle ABC is 30°.

The length of the opposite angle ABC is 0.5.

When θ is 90° then there isn’t a triangle anymore. The side that was opposite to the angle now points straight up and reaches the perimeter of the circle. The sin ratio is now one divided by 1. Sin θ = 1.

We have looked at the values 30°, 45°, 60° and 90° in the sin wave.

Now let’s look what happens as we increase the size of θ.

When θ is 120° then sin θ is 0.866 again. Remember, 0.866 = √3 / 2

What is the size of angle BAC when θ is 120°?

As the angle θ and angle BAC together lie on the x-axis which is a straight line, together they must add up to 180°. This means that angle BAC = 180 – 120 degrees which is 60°. sin 60 =√3 / 2 and sin 120 has the same value.