Consider a situation in which you are building a ramp so that people in wheelchairs can access a building. If the ramp must have a height of feet, and the angle of the ramp must be about , how long must the ramp be?
Solving this kind of problem requires trigonometry. Recall that in the first lesson, you learned that the word trigonometry comes from two words meaning triangle and measure. In this lesson we will define six trigonometric functions. For each of these functions, the elements of the domain are angles. We will define these functions in two ways: first, using right triangles, and second, using angles of rotation. Once we have defined these functions, we will be able to solve problems like the one above. (We will, in fact, solve such problems in lesson 7.)
The Sine, Cosine, and Tangent Functions
The first three trigonometric functions we will work with are the sine, cosine, and tangent functions. As noted above, the elements of the domains of these functions are angles. We can define these functions in terms of a right triangle: The elements of the range of the functions are particular ratios of sides of triangles.
We define the sine function as follows: For an acute angle in a right triangle, is the ratio of the side opposite of the angle to the hypotenuse of the triangle. For example, in the triangle shown above, we have:
Since all right triangles with the same acute angle are similar, this function is will produce the same ratio, no matter which triangle is used. Thus, it is a well defined function.
Similarly, the cosine of an angle is defined as the ratio of the side adjacent (next to) the angle to the hypotenuse of the triangle. In the triangle above, we have:
Finally, the tangent of an angle is defined as the ratio of the side opposite the angle to the side adjacent to the angle. In the triangle above, we have:
There are a few important things to note about the way we write these functions. First, keep in mind that the abbreviations and are just like . They simplify stand for specific kinds of functions. Second, be careful about how you pronounce the names of the functions. When we write it is still pronounced sine, with a long “”. When we write , we still say co-sine. And when we write , we still say tangent. (Sometimes casually people say “cos” and “tan, however, it shouldn’t be surprising that “sin” is always pronounced “sine”!)
We can use these definitions to find the sine, cosine, and tangent values for angles in a right triangle.
Example 1: Find the sine, cosine, and tangent of angle :
Solution:
One of the reasons that these functions will help us solve problems is that these ratios will always be the same, as long as the angles are the same. Consider for example, triangle similar to triangle .
If has length , then side of triangle is . Because is similar to , side has length . This means the hypotenuse has length . (We can show this either using the proportions from the similar triangles, or by using the Pythagorean Theorem.)
If we use triangle to find the sine, cosine, and tangent of angle , we get:
Example 2: Find using triangle and triangle
Solution:
Using triangle :
Using triangle :
Secant, Cosecant, and Cotangent functions
We can define three more functions also based on a right triangle.
Function name Definition Example
Secant In triangle ,
Cosecant In triangle ,
Cotangent In triangle ,
Example 3: Find the secant, cosecant, and cotangent of angle .
Solution:
First, we must find the length of the hypotenuse. We can do this using the Pythagorean Theorem:
Now we can find the secant, cosecant, and cotangent of angle :
Trigonometric Functions of Angles in Standard Position
Above, we defined the six trigonometric functions for angles in right triangles. We can also define the same functions in terms of angles of rotation. Consider an angle in standard position, whose terminal side intersects a circle of radius . We can think of the radius as the hypotenuse of a right triangle:
The point where the terminal side of the angle intersects the circle tells us the lengths of the two legs of the triangle. Now, we can define the trigonometric functions in terms of , , and :
Now we can extend these functions to include non-acute angles.
Example 4: The point is a point on the terminal side of an angle in standard position. Determine the values of the six trigonometric functions of the angle.
Solution:
Notice that the angle is more than , and that the terminal side of the angle lies in the second quadrant. This will influence the signs of the trigonometric functions.
Notice that the value of depends on the coordinates of the given point. You can always find the value of using the Pythagorean Theorem. However, often we look at angles in a circle with radius . As you will see next, doing this allows us to simplify the definitions of the functions.
The Unit Circle
Consider an angle in standard position, such that the point on the terminal side of the angle is a point on a circle with radius .
This circle is called the unit circle. With , we can define the trigonometric functions in the unit circle:
Notice that in the unit circle, the sine and cosine of an angle are the and coordinates of the point on the terminal side of the angle. Now we can find the values of the trigonometric functions of any angle of rotation, even the quadrantal angles, which are not angles in triangles.
We can use the figure above to determine values of the trig functions for the quadrantal angles. For example, .
Example 5: use the unit circle above to find each value:
a.
b.
c.
Solution:
a.
The ordered pair for this angle is . The cosine value is the coordinate, .
b. is undefined
The ordered pair for this angle is . The ratio is , which is undefined.
c.
The ordered pair for this angle is . The ratio is .
There are several important angles in the unit circle that you will work with extensively in your study of trigonometry: , , and . To find the values of the trigonometric functions of these angles, we need to know the ordered pairs. Let’s begin with .
The terminal side of the angle intersects the unit circle at the point . (You will prove this in one of the review exercises.). Therefore we can find the values of any of the trig functions of . For example, the cosine value is the coordinate, so . Because the coordinates are fractions, we have to do a bit more work in order to find the tangent value:
In the review exercises you will find the values of the remaining four trig functions of this angle. The table below summarizes the ordered pairs for , , and on the unit circle.
Angle coordinate coordinate
We can use these values to find the values of any of the six trig functions of these angles.
Example 6: Find the value of each function.
a.
b.
c.
Solution:
a.
The cosine value is the coordinate of the point.
b.
The sine value is the coordinate of the point.
c.
The tangent value is the ratio of the coordinate to the coordinate. Because the and coordinates are the same for this angle, the tangent ratio is .
Lesson Summary
In this chapter we have defined the six trigonometric functions. First we defined the functions for angles in right triangles, and then we defined them for angles of rotation. We considered angles formed when the terminal side of an angle intersected a circle of radius , and then we focused in on the unit circle, which has radius . The unit circle will be used extensively throughout the remainder of the chapter.
Points to Consider
How is the Pythagorean Theorem useful in trigonometry?
How can some values of the trig functions be negative
? How is it that some are undefined?
Why is the unit circle and the trig functions defined on it useful, even when the hypotenuses of triangles in the problem are not ?
Review Questions
Find the values of the six trig functions of angle .
Consider triangle VET below. Find the length of the hypotenuse.
Find the values of the six trig functions of angle .
The point is a point on the terminal side of an angle in standard position. Determine the radius of the circle.
Determine the values of the six trigonometric functions of the angle. The radius is .
The values are:
The point is a point on the terminal side of an angle in standard position. Determine the radius of the circle.
Determine the values of the six trigonometric functions of the angle. The radius is .
The values are:
The terminal side of the angle intersects the unit circle at . Use this ordered pair to find the six trig functions of .
In the lesson you learned that the terminal side of the angle intersects the unit circle at the point . Here you will prove that this is true.
Explain why Triangle is an equiangular triangle. What is the measure of angle ?
What is the length of ? How do you know?
What is the length of and ? How do you know?
Now explain why the ordered pair is .
Why does this tell you that the ordered pair for is ?
In the lesson you learned that the terminal side of the angle is . Use the figure below and the Pythagorean Theorem to show that this is true.
State the values of the six trig functions of .
In what quadrants will an angle in standard position have a positive tangent value? Explain your thinking.
Sketch the angle on the unit circle is. How is this angle related to ? What do you think the ordered pair is?
Review Answers
The length of the hypotenuse is .
The triangle is equiangular because all three angles measure . Angle measures because it is the sum of two degree angles.
has length because it is one side of an equiangular, and hence equilateral triangle.
and each have length , as they are each half of . This is the case because Triangle and are congruent.
We can use the Pythagorean theorem to show that the length of is . If we place angle as an angle in standard position, then and correspond to the and coordinates where the terminal side of the angle intersect the unit circle. Therefore the ordered pair is .
If we draw the angle in standard position, we will also obtain a triangle, but the side lengths will be interchanged. So the ordered pair for is .
Because the angle is in the first quadrant, the x and y coordinates are positive.
An angle in the first quadrant, as the tangent is the ratio of two positive numbers. An angle in the third quadrant, as the tangent in the ratio of two negative numbers, which will be positive.
The terminal side of the angle is a reflection of the terminal side of . From this, students should see that the ordered pair is .
Vocabulary
Adjacent
A side adjacent to an angle is the side next to the angle. In a right triangle, it is the leg that is next to the angle.
Hypotenuse
The hypotenuse is the longest side in a right triangle, opposite the right angle.
Leg
The legs of a right triangle are the two shorter sides.
Pythagorean Theorem
The Pythagorean theorem states the relationship among the sides of a right triangle:
Radius
The radius of a circle is the distance from the center of the circle to the edge. The radius defines the circle.
Unit Circle
The unit circle is the circle with radius and center . The equation of the unit circle is
Trigonometric Functions of Any Angle
Learning objectives
A student will be able to:
Identify the reference angles for angles in the unit circle.
Identify the ordered pair on the unit circle for angles whose reference angle is , , and , or a quadrantal angle, including negative angles, and angles whose measure is greater than .
Use these ordered pairs to determine values of trig functions of these angles.
Use tables and calculators to find values of trig functions of any angle.
Introduction
In the previous lesson we introduced the six trigonometric functions, and we worked with these functions in two ways: first, in right triangles, and second, for angles of rotation. In this lesson we will extend our work with trig functions of angles of rotation to any angle in the unit circle, including negative angles, and angles greater than . In the previous lesson, we worked with the quadrantal angles, and with the angles , , and . In this lesson we will work with angles related to these angles, as well as other angles in the unit circle. One of the key ideas of this lesson is that angles may share the same trig values. This idea will be developed throughout the lesson.
Reference Angles and Angles in the Unit Circle
In the previous lesson, one of the review questions asked you to consider the angle . If we graph this angle in standard position, we see that the terminal side of this angle is a reflection of the terminal side of , across the axis.
Notice that makes a angle with the negative axis. Therefore we say that is the reference angle for . Formally, the reference angle of an angle in standard position is the angle formed with the closest portion of the axis. Notice that is the reference angle for many angles. For example, it is the reference angle for and for .
In general, identifying the reference angle for an angle will help you determine the values of the trig functions of the angle.
Example 1: Graph each angle and identify its reference angle.
a.
b.
c.
Solution:
a. makes a angle with the axis. Therefore the reference angle is .
b. makes a with the axis. Therefore the reference angle is .
c. is a full rotation of , plus an additional . So this angle is co-terminal with , and is its reference angle.
If an angle has a reference angle of , , or , we can identify its ordered pair on the unit circle, and so we can find the values of the six trig functions of that angle. For example, above we stated that has a reference angle of . Because of its relationship to , the ordered pair for is is . Now we can find the values of the six trig functions of :
Example 2: Find the ordered pair for and use it to find the value of .
Solution:
As we found in example 1, the reference angle for is . The figure below shows and the three other angles in the unit circle that have as a reference angle.
The terminal side of the angle represents a reflection of the terminal side of over both axes. So the coordinates of the point are . The coordinate is the sine value, so .
Just as the figure above shows and three related angles, we can make similar graphs for and .
Knowing these ordered pairs will help you find the value of any of the trig functions for these angles.
Example 3: Find the value of
Solution:
Using the graph above, you will find that the ordered pair is . Therefore the cotangent value is
We can also use the concept of a reference angle and the ordered pairs we have identified to determine the values of the trig functions for other angles.
Trigonometric Functions of Negative Angles
Recall that graphing a negative angle means rotating clockwise. The graph below shows .
Notice that this angle is coterminal with . So the ordered pair is . We can use this ordered pair to find the values of any of the trig functions of . For example, .
In general, if a negative angle has a reference angle of ,, or , or if it is a quadrantal angle, we can find its ordered pair, and so we can determine the values of any of the
trig functions of the angle.
Example 4: Find the value of each expression.
a.
b.
c.
Solution:
a.
is in the quadrant, and has a reference angle of . That is, this angle is coterminal with . Therefore the ordered pair is and the sine value is .
b.
The angle is in the quadrant and has a reference angle of . That is, this angle is coterminal with . Therefore the ordered pair is and the secant value is .
c.
The angle is coterminal with . Therefore the ordered pair is and the cosine value is .
We can also use our knowledge of reference angles and ordered pairs to find the values of trig functions of angles with measure greater than .
Trigonometric Functions of Angles Greater than 360 Degrees
Consider the angle . As you learned previously, you can think of this angle as a full degree rotation, plus an additional . Therefore is coterminal with . As you saw above with negative angles, this means that has the same ordered pair as , and so it has the same trig values. For example,
CK-12 Trigonometry Page 4