In this explainer, we will learn how to find reference angles and how to use them to find the values of trigonometric functions.
We recall that we can evaluate trigonometric functions by sketching the argument in standard position and then determining the coordinates of the point of intersection between the terminal side of the argument and the unit circle centered at the origin. To sketch an angle in standard position, we measure in the counterclockwise direction from the positive -axis when the angle is positive and in the clockwise direction if the angle is negative.
For example, we can find the value of from the following diagram.
The coordinates of the point of intersection between the unit circle centered at the origin and the terminal side of the angle in standard position are . We can evaluate the trigonometric expression for each coordinate by using the diagram and trigonometry. First, we note that the angles in a straight line sum to , so we can add the angle to the diagram as follows.
Second, since the unit circle centered at the origin is the locus of all points a distance of 1 from the origin, we know that the line segment between and the origin is of length 1. If we drop a perpendicular from the point to the -axis, we get the following right triangle.
We use the absolute value of the coordinates of the point since we want the lengths of the sides of the triangle rather than the coordinates. Finally, we can determine these values using trigonometry; the sine of an angle is the ratio between the lengths of the side opposite the angle and the hypotenuse. Hence,
Similarly, the cosine of an angle is the ratio between the lengths of the side adjacent to the angle and the hypotenuse. Hence,
We know that
Therefore, by considering the fact that the point lies in the second quadrant, we have
In a similar manner, we can evaluate trigonometric values of angles beyond . For example, we can evaluate by sketching the angle in standard position and noting it is the same as the angle in standard position.
Since the terminal side of an angle in standard position is invariant under full revolutions clockwise or counterclockwise, this means that the sine and cosine of the angle are periodic about . We can then evaluate by finding the measure of the angle between the terminal side and the positive -axis; in this case, we note that a full revolution is , so its measure is . Adding this to the diagram and the unit circle centered at the origin gives us the following.
We can then determine the value of by dropping a perpendicular to the -axis and applying trigonometry.
Since the point lies in the fourth quadrant, we see that is negative. Hence, by applying trigonometry to the right triangle, we have
In the above examples, we were able to evaluate trigonometric functions for any angle by first finding an equivalent positive angle in standard position and then determining the measure of the acute angle the terminal side makes with -axis. We call the equivalent positive angle the principal angle, and the measure of the acute angle that the terminal side makes with the -axis the reference angle; we define these formally as follows.
Definition: Principal Angle
If is an angle in standard position, then the counterclockwise angle between the initial and terminal side of (less than a full turn) is called the principal angle of .
Definition: Reference Angle
If is an angle in standard position, and not a quadrantal angle (integer multiple of a right angle), then the measure of the acute angle the terminal side makes with the -axis is called the reference angle of .
There are four different possibilities for the principal and reference angles based on which of the four quadrants the terminal side lies in, which we can see below.
Let’s see some examples of how to determine the principal angle of various angles given in radians.
Example 1: Identifying the Principal Angle for a Negative Angle
Given the angle , find the principal angle.
Answer
We recall that to find the principal angle of , we sketch the angle in standard position and then find the counterclockwise angle between the initial and terminal side, which is less than a full turn, . To sketch in standard position, we note the value is negative, so we measure the angle in a clockwise direction from the positive -axis to give us the following, where we label the principal angle .
While the directed angle is negative, if we take the magnitude of this angle, then together with we have a full angle of , giving us
Hence, the principal angle of is .
Example 2: Identifying the Principal Angle for an Angle Greater than 2𝜋
Given the angle , find the principal angle.
Answer
We recall that to find the principal angle of , we sketch the angle in standard position and then find the counterclockwise angle between the initial and terminal side, which is less than a full turn.
To sketch in standard position, we note that , so it represents more than a full turn. This means we need to remove integer multiples of to find an equivalent angle in standard position. Since , we subtract as follows:
We can see that and have the same terminal angle when drawn in standard position in the following diagrams.
Since this value is between 0 and , we can conclude the principal angle of is .
Let’s now see an example of how to use the principal value to evaluate a trigonometric expression without a calculator.
Example 3: Evaluating the Cosine of a Negative Angle by Finding the Principal Angle
Find without using a calculator.
Answer
We recall that we can evaluate trigonometric functions by sketching angles in standard position and then determining the coordinates of the intersection between the terminal side of the angle and the unit circle centered at the origin. We could do this by sketching the angle in standard position. However, we can also find the principal angle as this will give the same terminal side.
The principal angle of will be between and , and the terminal side of the angle in standard position will be the same. We can find the principal angle by adding integer multiples of to the angle. When we do, we obtain
We can then determine the value of by sketching in standard position along with the unit circle centered at the origin. The -coordinate of the point of intersection between the circle and terminal side will be .
To determine the -coordinate of this point, we note that the angles in a straight line sum to , so the angle between the negative -axis and the terminal side measures
Dropping a perpendicular from the point of intersection to the -axis then gives us the following.
By applying right triangle trigonometry, the side adjacent to the angle will have a length of .
Since the point of intersection has a negative -coordinate, we see that
In the above example, we evaluated a trigonometric expression by finding the principal angle of its argument. In actual fact, we also used the reference angle of this principal value.
Let’s see another example of this.
Example 4: Evaluating the Sine of an Angle by Finding the Reference Angle
Find the value of .
Answer
We recall that we can evaluate trigonometric functions by sketching the argument in standard position and then determining the coordinates of the intersection between the terminal side of the angle and the unit circle. We start by sketching in standard position, noting that it is positive, so the angle is measured counterclockwise from the positive -axis. Since , the terminal side will lie in the fourth quadrant, as shown in the following diagram.
To determine the value of , we need to find the -coordinate of the point of intersection. We do this by finding the reference angle of ; that is, the measure of the acute angle between the terminal side and the -axis when is drawn in standard position. We can see in the diagram that the angle between the terminal side and the -axis and add together to make . Labeling the reference angle as , we have
We then add this to our diagram and drop a perpendicular from the intersection point to the -axis, as shown in the following diagram.
We can add a direction to the reference angle to see that this is the angle in standard position. The coordinates of the point of intersection can also be written as . Equating the two expressions for the -coordinate of the point of intersection and evaluating, we have
Hence, .
In our next example, we will use the reference angle of an argument to evaluate a reciprocal trigonometric function.
Example 5: Evaluating the Secant of an Angle by Finding the Reference Angle
Find without using a calculator.
Answer
To determine the secant of an angle, we first need to recall that the secant function is the reciprocal of the cosine function. This gives us
We can determine the cosine of by sketching the angle in standard position and then finding the -coordinate of the point of intersection between the terminal side of the angle and the unit circle. Since is positive, the angle is measured in a counterclockwise direction, and we note that , so the terminal side will lie in the fourth quadrant. This gives us the following sketch.
To determine the -coordinate of the point of intersection, we find the reference angle for , which is the measure of the acute angle between the terminal side and the -axis in the diagram above. Labeling the reference angle and noting that these two angles make a full revolution, we have
Adding this angle into our diagram and dropping a perpendicular from the point of intersection to the -axis gives us the following.
Since the terminal side lies in the fourth quadrant, the -coordinate of the point of intersection is positive, which means that is also positive. We can determine the exact value by applying trigonometry to the right triangle in the diagram; the side adjacent to the angle has a length of , and the hypotenuse has a length of 1.
The cosine of is then the ratio of the lengths of the side adjacent to the angle and the hypotenuse, giving us
Finally, we take the reciprocal of both sides of this equation to see that
In our final example, we will evaluate the tangent function by first finding the reference angle of its argument.
Example 6: Evaluating the Tangent of an Angle by Finding the Reference Angle
Find the exact value of without using a calculator.
Answer
To determine the tangent of an angle without a calculator, we first recall that the tangent is the quotient of the sine and cosine of the same angle. Applying this to the angle gives us
We can determine the sine and cosine of an angle by sketching it in standard position and then finding the coordinates of the point of intersection between the terminal side of the angle and the unit circle centered at the origin. Since is positive, the angle is measured in a counterclockwise direction, and we note that . So, the terminal side will lie in the third quadrant. This gives us the following sketch.
To determine the values of the - and -coordinates of the point of intersection, we need to find the measure of the angle between the terminal side and the -axis (called the reference angle). The angle between the positive -axis and negative -axis is , so the reference angle is
We can add this to our diagram and drop the perpendicular from the point of intersection to the -axis to give us the following.
Since the point of intersection is in the third quadrant, both the - and -coordinates of the point of intersection will be negative. Therefore, the base of the right triangle will have a length of and a height of , giving us the following right triangle.
We can find expressions for and by applying trigonometry to the right triangle. First, by taking the ratio of the lengths of the side opposite to and the hypotenuse, we have
Second, by taking the ratio of the lengths of the side adjacent to and the hypotenuse, we have
Finally, we can take the quotient of these values to find the tangent of the argument as follows:
Hence,
Let’s finish by recapping some of the important points from this explainer.
Key Points
- If is an angle in standard position, then the counterclockwise angle between the initial and terminal side of (less than a full turn) is call the principal angle.
- Taking a principal value does not change the value of the sine or cosine functions.
- If is an angle in standard position, and not a quadrantal angle, then the measure of the acute angle the terminal side makes with the -axis is called the reference angle.
- By sketching an angle in standard position and using the principal and reference angles, we can find equivalent arguments to evaluate trigonometric functions.
