Fundamentals of Unit Circle Trigonometry

Image by Arek Socha from Pixabay.

Diagrams, created by the author, are to teach concepts. These hand drawn diagrams are licensed under the Creative Commons Attribution-Share Alike 4.0 International license. The drawings were created using Google Jamboard. All images that came from a different source, not the author, have a caption with the credits. Diagrams will provide a great deal of the instruction, so look at them carefully. This will be great for visual learners.

A right triangle

Parts of a triangle

Tangent

This is not too important for this specific article.

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Sine

Cosine

Radians

A measure of angle size. One radian is equal to the angle formed between two line segments drawn from the center of a circle to the ends of an arc equal to the length of the radius of the circle.A measure of angle size. One radian is equal to the angle formed between two line segments drawn from the center of a circle to the ends of an arc equal to the length of the radius of the circle.

Sine on a unit circle

Sine(theta) is actually the y coordinate on the circle. The triangle derives that and allows us to break away from the confines of an embedded triangle. Here, the length of the opposite side is sin(theta) because the sin(theta)=opposite/hypotenuse. The hypotenuse is one, so the sin(theta)=opposite/one=opposite.

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Cosine on a unit circle

Cosine is actually the x coordinate on the circle. The triangle derives that and allows us to break away. Again, since this is a unit circle with a hypotenuse of one, cos(theta)=adjacent.

So, in the context of the coordinates of a point on the edge of a circle:

cos(Θ)=x

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sin(Θ)=y

Theta is the symbol that looks like a zero with a crossbar.

Theta

In the context of a circle, theta is the angle formed between the line segment running from the circle center to the point on the circle in question (that we are finding the x or y value of) and the positive x or y axis of the circle when going in a counter-clockwise direction.

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Why does the sine graph look the way it does?

This is a graph of the length of the opposite side of every possible right triangle embedded into the circle that touches a point on the edge, the middle, and the x axis! Or, this is a graph of the angle in radians of theta on the x axis vs. the y coordinate value of the corresponding points on a circle (the distance of a point on the edge of a circle from the x axis).

Also known as sin(Θ). Here is a graph of sin(x):

Graph courtesy of Desmos.

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Why does the sine graph repeat?

The sine graph repeats over and over again as we move side to side on the horizontal axis. This is because the angle of theta in radians is on the x-axis. If you start at the positive x axis on a circle and keep drawing line segments from the circle’s middle to a point on the edge, you can keep going around the circle in a counterclockwise direction forever. When we go around the circle once and get to the positive x axis again, theta is 2pi radians. We can go around again and get to 4pi and so on.

Why does the graph never exceed 1 or get below -1 on the y axis?

It is important to note that this is a graph of a unit circle. So, the hypotenuse of the unit circle is equal to one. Also, the y coordinate of a point on the edge of a circle is what is being plotted on the y axis of this sine graph. The y coordinate of a point on the edge of a circle can never exceed 1 or get below -1. This is because the y coordinate is really the distance from the circle’s x-axis.

At the middle of the circle, where the y axis is located, the edge of the circle is one radius length away from the x axis. This means that the distance from the x axis is 1. That is the peak. Then the distance starts decreasing again. It is is the exact same story directly at the bottom of the circle, at the negative y axis. The sine of 90 degrees is 1. Any other angle will get less than one. The sine of 270 is -1. Any other angle will get greater than one.

Why does the sine graph NOT look like this?

It does not look like this because this would assume that y increases and decreases linearly towards 1 and away from 1 (and the same for -1). This is the shape that would match the jagged graph. It is very linear:

But, look at this circle:

With the circle, the distance of a point on the edge of the circle from the x axis (the y coordinate value) initially increases fast but then slows down. This causes the sine curve. At the beginning of the sine cycle, the slope is greater. But then it decreases as it approaches y=the radius of the circle. This matches what can be seen on the circle: the initial rapid movement away from the x axis that tapers off to be less and less as y approaches the radius. That is why the sine graph is curved instead of jagged.

Why is the length of one cycle 2pi radians?

The angle of an entire circle, 360 degrees, is 2pi radians. Why?

One cycle of a sine graph (one up and one down) is one trip around a circle. A cycle is just the smallest part of the graph that can be repeated many times to form the entire graph. After one cycle is finished the next begins. The next one is identical. This is mirrored in the circle because after going 2pi radians around the circle, the starting point is arrived at again and the whole thing begins again. Just like circular clocks keep repeating themselves after they go around once, a cycle (twelve hours).

One sine cycle:

An angle measuring 2pi radians:

One way to solve a problem using a sine graph:

It can be used to find the y coordinate of a point on a circle.

Another way to solve a problem using a sine graph:

The graph can be used to find sine ratios for theta measured in radians. This is because on a sine graph theta is on the horizontal axis and the sine ratio is on the vertical axis.

Point (1.57080321,1) falls on the graph of sin(x). This means that the sin(1.57080321 radians)=1/1

Graph courtesy of Google.

Point (3.00197996, 0.139159591) falls on the graph of sin(x). This means that the sin(3.00197996 radians)=0.139149591/1

Graph courtesy of Google.

Why does the cosine graph have the shape that it does?

Cosine is actually the x coordinate of a point on the edge of a unit circle, or the distance from the y axis. This is the opposite of sine. Cosine is like sine rotated ninety degrees. But since circles can be rotated any amount without changing in any way, the profile, the shape of the cosine graph is no different than the sine graph. The shape of a cosine graph is the same as a sine graph but shifted over. All the reasoning for the sine graph is the same for cosine, with the only exception being that the x coordinate is on a cosine graph’s vertical axis. Here is a graph of cos(x):

Graph courtesy of Desmos.

Cos(x), red, and sin(x), blue, superimposed:

So, why is cos(x) shifted over by ½ pi radians?

Cos(x) is shifted over because the cosine of theta is equal to the x coordinate value of the corresponding point on the edge of the circle, the distance from the y axis. The sin(0) is equal to 0 because theta is the angle between the line segment drawn from the point on the edge of the circle to the middle and the positive x axis. This causes the graph of the sin(x) to start at the point when it is closest to the x axis and increase up to 1 as the point on the edge of the circle gets farther from the x axis.

But the cos(0) is 1. This is because when theta is 0, the point at the edge of the circle is at the x axis, that point is the maximum distance away positively from the y axis, at 1 on a unit circle. This causes the cos(0) to be 1 and the graph of the cos(x) to start at 1 and decrease from there as the point approaches the y axis. Since the cosine graph starts at 1, its maximum, and the sine graph starts at 0, its middle point, the two are not in sync by ¼ of a cycle. This causes them to be shifted over by ½ pi radians. 

This illustrates why cosine is just shifted over:

cos(0)=1

sin(0)=0

The End

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