Unit Circle Trigonometry

Trigonometry on the Unit Circle

The unit circle demonstrates the output of the trigonometric functions: sine, cosine, tangent and cotangent. A point on the perimeter of the circle forms an inscribed right-triangle of hypotenuse length . The side lengths of this triangle and the line tangent to the point on the circle correspond to the output of the trigonometric functions[1].

Summary

Shown below is a table summarizing the main functions and the lengths they correspond to.

Geometry Function
Geometry of the Sine Function The output of the sine function corresponds to the distance from a point on the perimeter of the circle to the x-axis. This length is the same as the opposite side of the right-triangle formed by the point.
Geometry of the Cosine Function The output of the cosine function corresponds to the distance from a point on the perimeter of the circle to the y-axis. This length is the same as the adjacent side of the right-triangle formed by the point.
Geometry of the Tangent Function The output of tangent corresponds to the length of the line tangent to a point on the circle starting from the point and intersecting with the x-axis.
Geometry of the Cotangent Function The output of cotangent corresponds to the length of the line tangent to a point on the circle starting from the point and intersecting with the y-axis.

Background

The unit circle is a circle of radius placed at the origin of the coordinate system. For many students, a fair objection when introduced to the unit circle is that trigonometry is about triangles. However, since each of the trigonometric functions return the ratio of two sides of a right triangle and each of the functions take an angle as input, it can be argued that trigonometry is really about circles.

Unit Circle Point

First, let’s take a look at the definitions of sine and cosine. The functions return the ratio of one of the sides over the hypotenuse of a right triangle as shown in their definitions below:

Right Triangle

Since the side length of a right-triangle is always less than or equal the magnitude (absolute length) of the hypotenuse, the output of the functions will always be in the range of . In a related fashion, if the hypotenuse is constrained to the length of a right-triangle placed on the unit circle describes all possible output of the sine and cosine function. Click and drag the blue control point on the right-triangle placed on the unit circle below:

Next, as well as being defined by coordinates, a point on the unit circle is described by and angle (theta). The trigonometric functions bridge these two different ways of defining a point. Note, mathematicians prefer to use the radians angle system to describe angles, which measures an angle using the length of the radius of the circle. A point defining an angle is shown in the interactive below:

The trigonometric functions are periodic, meaning that the output continues to repeat forever. This can be connected to the nature of an angle. If you walk along the circumference of the cicle one full rotation equal to (tau) radians and then continue walking, the total distance traveled will be more than a full rotation. However, the measured angle will be smaller and can be described as a fraction of the circle’s circumference.

Sine Function

Shown below is the graph of the sine function and the right-triangle placed on the unit circle. The input angle is highlighted in red and the output corresponding to the opposite length of the right triangle is highlighted in blue. Click and drag either control point to see the input and output of the function.

Cosine Function

Shown below is the graph of the cosine function and the right-triangle placed on the unit circle. The input angle is highlighted in red and the output corresponding to the adjacent length of the right triangle is highlighted in blue. Click and drag either control point to see the input and output of the function.

Tangent Function

Shown below the output of the tangent function correspond to the line tangent to the point on the unit circle defined by the point on the unit circle and the point where the line intersects the x-axis.

Tangent Function Geometry

Cotangent Function

Shown below the output of the cotangent function correspond to the line tangent to the point on the unit circle defined by the point on the unit circle and the point where the line intersects the y-axis.

Coangent Function Geometry

Good Links

Sine Function | Function

The sine function returns the sine of a number provided in radians. In geometric terms, given the angle of a right-triangle as input, the function returns the ratio of the triangle's opposite side over its hypotenuse.

Cosine Function | Function

The cosine function returns the cosine of an angle provided in radians. In geometric terms, the function returns the ratio of the right-triangle's adjacent side over its hypotenuse.

Tangent Function | Function

Given the angle of a right triangle as input returns the ratio of the triangle's opposite side over its adjacent side.

Cotangent Fuction | Function

Given the angle of a right triangle as input, returns the ratio of the adjecent side over the opposite side.

Radians | Concept

Radians are a unit that measure angle using the radius of a circle. One radian is equal to the amount of rotation required to travel the length of one radius along the circumference of the circle.

References

  1. Unit Circle Trigonometry | Figure