In this explainer, we will learn how to relate the - and -coordinates of points on the unit circle to trigonometric functions.
The unit circle is a circle with a radius of 1 whose center lies at the origin of a coordinate plane. For any point on the unit circle, a right triangle can be formed as in the following diagram. The hypotenuse of this right triangle makes an angle with the positive -axis.
Using right triangle trigonometry, we can define the trigonometric functions in terms of the unit circle:
We note that is not defined when . We also observe that, while we have derived these definitions for an angle in quadrant 1, they hold for an angle in any quadrant.
Theorem: Trigonometric Functions and the Unit Circle
The - and -coordinates of a point on the unit circle given by an angle are defined by
In our first example, we will demonstrate how we can use these definitions of trigonometric functions in the unit circle to find exact values, given information about the terminal side of an angle.
Example 1: Finding the Value of a Trigonometric Function of an Angle given the Coordinates of the Point of Intersection of the Terminal Side and the Unit Circle
Find , given that is in standard position and its terminal side passes through the point .
Answer
An angle is said to be in standard position if the vertex is at the origin and the initial side lies on the positive -axis. The angle is measured in a counterclockwise direction from the initial side to the terminal side. Hence, the angle is as shown.
We draw a right triangle with side lengths of units and units as demonstrated below.
Now, we can use the Pythagorean theorem to calculate the value of the missing dimension in the triangle:
Since , . This tells us that the point lies on the unit circle. We recall that the - and -coordinates of a point on the unit circle given by an angle are defined by
is therefore equal to the value of the -coordinate of the point, which is :
In addition to the standard trigonometric functions, it is also possible to define the reciprocal trigonometric functions (the reciprocal of a number is ). We can define them as follows.
Definition: Reciprocal Trigonometric Functions
For an angle , the reciprocal trigonometric functions are as follows:
- The cosecant function: , for
- The secant function: , for
- The cotangent function: , for
Since we can write the standard trigonometric function in terms of the unit circle, it is also possible for us to write the reciprocal functions in terms of the unit circle. That is, let us once more consider a point on the unit circle, with angle to the positive -axis.
Then, the reciprocal trigonometric functions can be written as follows:
Let us consider an example where we use the unit circle to find the exact value of the secant function.
Example 2: Finding the Value of a Trigonometric Function of an Angle given the Coordinates of the Point of Intersection of the Terminal Side and the Unit Circle
Find , given that is in standard position and its terminal side passes through the point .
Answer
An angle is said to be in standard position if the vertex is at the origin and the initial side lies on the positive -axis. The angle is measured in a counterclockwise direction from the initial side to the terminal side. Hence, the angle is as shown.
To calculate the value of , we will begin by determining whether the point with coordinates lies on the unit circle. To do this, we can consider a right triangle with side lengths of units and units as shown below.
Then, we can calculate the length of the hypotenuse, , by using the Pythagorean theorem:
Therefore, , which means . So, we have shown that the point lies on the unit circle.
The - and -coordinates of a point on the unit circle given by an angle are defined by
We recall that . In the case of the unit circle, .
We will now demonstrate how to find the exact value of a quadrantal angle, that is, an angle whose terminal side lies on either the - or -axis.
Example 3: Finding Cosine Values of Quadrantal Angles
Find the value of .
Answer
The - and -coordinates of a point on the unit circle given by an angle are defined by
The value of will therefore be the value of the -coordinate at the point where the terminal side for intersects the circumference of the unit circle, as shown on the diagram.
The terminal side of the angle lies on the -axis, so the point at which it intersects the unit circle is . The -coordinate, and hence the value of , is 1.
The value of is 1.
In our previous examples, we showed how to use the definition of the unit circle to find the exact value of trigonometric functions. In our next example, we will demonstrate how the unit circle validates the periodicity of such functions.
Example 4: Exploring the Different Angles between 0 and 2𝜋 That Have the Same Trigonometric Function
Suppose is a point on a unit circle corresponding to the angle of . Is there another point on the unit circle representing an angle in the interval that has the same tangent value? If yes, give the angle.
Answer
We will begin by sketching the unit circle and the point which creates an angle of radians with the positive -axis measured in a counterclockwise direction. Since is between and , we know that this point must lie in the third quadrant. Since both the - and -coordinates of this point are negative, we will define it as for some positive constants and .
We now recall the definition of the tangent function with respect to the unit circle. Given an angle in standard position, where the coordinates of the point of intersection of the terminal side with the unit circle are and ,
For point ,
The quotient of these - and -coordinates is positive. We now observe that there must be a second point on the unit circle for which this is the case. This is the point with coordinates which lies in the first quadrant.
We can map point onto point by performing a single rotation of radians. To find the value of such that ,
Yes, there is another point on the unit circle that produces the same tangent value as the angle . The angle is .
In our final example, we will demonstrate how to use the unit circle to evaluate a simple trigonometric function.
Example 5: Finding the Value of a Trigonometric Function given the Coordinates of the Point of Intersection of the Unit Circle with the Terminal Side of an Angle in Standard Position
The terminal side of in standard position intersects the unit circle at the point with coordinates , where . Find the value of .
Answer
An angle is said to be in standard position if the vertex is at the origin and the initial side lies on the positive -axis. Since is in standard position and is not on the -axis, the point must lie on the positive -axis. We can therefore sketch on the unit circle. Since both - and -coordinates are positive, point lies in the first quadrant.
We know that the - and -coordinates of a point on the unit circle given by an angle are defined by
The value of is therefore equal to the value of the -coordinate of point . By representing as a right triangle, we can find the value of by using the Pythagorean theorem.
We have
Since in this example, we obtain units.
The value of is .
Let us finish by recapping some key concepts from this explainer.
Key Points
- The unit circle is a circle with a radius of 1 centered at the origin of coordinate plane.
- The - and -coordinates of a point on the unit circle given by an angle are defined by
- The tangent ratio can also be defined for points on the unit circle , where :
- The CAST diagram can be used to identify the signs of trigonometric function for angles in each quadrant.
