Module 2: Basic Concepts of Trigonometric Functions

Topics

1. Right-Triangle Trigonometry
   1. Trigonometric Ratios
   2. Relationships between Trigonometric Functions
   3. Trigonometric Function Values of 30°, 45°, and 60° Angles
   4. Using a Calculator
   5. Solving Right Triangles
   6. Applications of Right Triangles
2. Trigonometric Functions of Any Angle
   1. Angles in Standard Position
   2. Reference Triangles and Trigonometric Function Values
   3. Trigonometric Function Values of Multiples of 90°
   4. Symmetry and Reference Angles
3. The Unit Circle and Angle Conversion
   1. Radian Measure
   2. Linear Speed and Angular Speed
4. Circular Functions and Their Graphs
   1. Definitions of Circular Functions
   2. Graphs of the Sine and Cosine Functions
   3. Graphs of the Other Trigonometric Functions
5. Transformations of Sine and Cosine Functions

I. Right-Triangle Trigonometry

After completing this section, you should be able to:

• find the six trigonometric function values of any acute angle of a right triangle
• determine the exact trigonometric function values of 30°, 45°, and 60° angles
• use a calculator to find trigonometric function values
• solve any right triangle by calculating its side and angle measurements
• use trigonometry to solve applied problems related to right triangles

A. Trigonometric Ratios

The study of trigonometry begins by focusing upon the relationships between the sides and angles of right triangles.

First consider a brief review of terminology.
A triangle is a three-sided polygon.
Each pair of sides of a triangle has a common point called a vertex.
Each pair of sides forms an angle interior to the triangle.

If a disk were cut into 360 equal wedges, each wedge would contain an angle of 1 degree. This idea of using 360 divisions originated with the ancient Babylonians, and it remains embedded in the current degree system of angle measurement.

Recall that for any triangle, the sum of the measures of the three angles is 180°.

A right triangle is a triangle that contains a right angle. In a right triangle, since the sum of the three angles is 180° and one of the angles is 90°, the sum of the measures of the two remaining angles must be 180° – 90°, or 90°. Therefore, the two remaining angles in a right triangle are both acute and complementary.

The ratios of the lengths of the sides are called trigonometric ratios, or trigonometric function values, of the angle θ. There are six ratios.

One of the ratios is the length of side opposite θ divided by the length of the hypotenuse. This ratio is called the sine of θ.

The other five ratios are called the cosine, tangent, cosecant, secant, and cotangent. The ratios are defined as follows:

B. Relationships between Trigonometric Functions

From the definitions and previous examples, it is easy to see that there are relationships among the trigonometric function values.

If you know the sine, cosine, and tangent of an angle, then by taking the reciprocals, you get the cosecant, secant, and cotangent.

There is a mnemonic aid that can help you to remember the definitions for the sine, cosine, and tangent ratios. Think of Professor SOH-CAH-TOA (pronounced phonetically as "soak a toe-a"). The joke is that the Professor is a wizard at trigonometry, but has a sore toe which needs to be soaked!

For the example involving the 3-4-5 triangle, notice that the trigonometric ratios of angle θ are related to the trigonometric ratios of angle . For example, sin θ = 3/5 = cos and cos θ = 4/5 = sin . Since angles θ and are complementary, = 90° – θ.

sin θ = cos = cos (90° – θ) and cos θ = sin = sin (90° – θ).

Because of this relationship, the sine and cosine functions are called cofunctions. There are similar relationships between the tangent and cotangent, and between the secant and cosecant functions. These relationships are listed below.

An interesting question to ask is this: For right triangles having the same proportions, does a larger triangle have larger trigonometric function values? To answer this, consider similar triangles.

Recall that two triangles are similar if the corresponding angles have the same measure or—equivalently—the corresponding sides are proportional. For example, the 3-4-5 triangle and the 6-8-10 triangle are similar. Each side in the 6-8-10 triangle is twice as long as the corresponding side in the 3-4-5 triangle.

For the 6-8-10 right triangle, the set of six trigonometric ratios associated with angle θ are given by

The 6-8-10 triangle has the same trigonometric ratios as the 3-4-5 triangle.

An analogous result applies to any two similar triangles.

Given just one of the trigonometric ratios, it is possible to use the Pythagorean theorem to assist in determining the remaining five trigonometric ratios.

C. Trigonometric Function Values of 30°, 45°, and 60° Angles

There are certain acute angles that appear very frequently in trigonometric applications: 30°, 45°, and 60°. It is worthwhile to compile a table of the trigonometric function values for these angles. Fortunately, it is relatively simple to derive the values, by using basic geometric facts and working with appropriate triangles.

Recall that an isosceles triangle is a triangle having two sides of the same length or—equivalently—two angles of the same measure.

For an isosceles right triangle, both acute angles must have measures of 45°, since the acute angles are complementary and have equal measure.

Take a square and cut it in half along a diagonal to get two isosceles right triangles. Pick one of the isosceles right triangles and label the legs and the hypotenuse.

For convenience, suppose the legs each have length 1. Use the Pythagorean theorem to find the length of the hypotenuse:

Since 1² + 1² = c², then 2 = c² and the length of the hypotenuse is.

Now it is easy to determine the trigonometric function values of an angle of 45°.

Next, determine the trigonometric function values of 30° and 60°. Notice that if a right triangle contains an angle of 60°, then the other acute angle must have a measure of 30°, since the angles are complementary.

An equilateral triangle is a triangle having all three sides of the same length, or equivalently all three angles of the same measure. Because the sum of the measures of the three angles must be 180°, each angle in an equilateral triangle must measure 60°.

Take an equilateral triangle and cut it in half to get two right triangles.

Each right triangle has acute angles 30° and 60°.

For convenience, suppose that the sides of the equilateral triangle have length 2. Then each right triangle has a hypotenuse of length 2 and a leg of length 1.

Use the Pythagorean theorem to find the length of the other leg:

1² + b² = 2², so b² = 3 and the length of the other leg is.

Now it is easy to determine the trigonometric function values for angles of 30° and 60°.

The sine, cosine, and tangent function values of 30°, 45°, and 60° are summarized in the following table.

For the 45° angle, it is good to keep in mind the 1-1- triangle.

For the 30° and 60° angles, it is good to keep in mind the 1--2 triangle. The shortest side is the side opposite the 30° angle.

By drawing and referring to the appropriate triangle, you can quickly derive the trigonometric function values without consulting the table.

D. Using a Calculator

Trigonometric function values of any angle can be found using a scientific or graphing calculator. Most calculators find values for the sine, cosine, and tangent, and the user finds the cosecant, secant, and cotangent by computing the appropriate reciprocal.

As you will see later on in topic III-A, there is more than one unit of measurement for angles. Be sure to put your calculator in degree mode (denoted DEG) when working with angles measured in degrees.

Recall that very large numbers and very small numbers are often written in scientific notation. Many calculators display a letter E preceding the exponent. For example, a calculator display of 1.6520000E-03 represents 1.6520000 × 10^-3 = 0.001652.

So far, you have taken an angle and found a corresponding trigonometric value. Going from a trigonometric value to an angle is the inverse process. (Inverse trigonometric functions will be discussed in detail in module 3.)

A calculator can be used to carry out the inverse process, to find an acute angle that has a given trigonometric value. Typically, a calculator uses a SHIFT key or an INV key. Put your calculator into degree (DEG) mode. Enter the trigonometric value, press SHIFT (or INV), then the trigonometric function. (You may need to consult the instructions for your calculator.)

E. Solving Right Triangles

To solve a triangle is to determine the lengths of all three sides and the measures of all three angles. Solving a triangle is similar to solving a puzzle. There are six values to find.

In the case of a right triangle, one of the angles must have a measure of 90°, and the lengths of the sides are related by the Pythagorean theorem. So, with just a little more information, it is possible to solve the right triangle.

Problem 2: Given a right triangle, suppose you know the measure of one of the acute angles and the length of one side. That is, you know one angle and the length of either the hypotenuse or the side opposite the angle or the side adjacent to the angle.

F. Applications of Right Triangles

The branch of mathematics known as trigonometry arose from the need to solve real-world problems involving right triangles.

Applications often involve a horizontal line of reference. An angle measured above the horizontal line is called an angle of elevation. An angle measured below the horizontal line is called an angle of depression.

II. Trigonometric Functions of Any Angle

After completing this section, you should be able to:

• determine the trigonometric function values of any angle in standard position
• use reference triangles, symmetry principles, and reference angles as aids in finding trigonometric function values

A. Angles in Standard Position

The notion of a trigonometric function can be extended to angles which are not acute angles. To describe a general angle, the angle is put in a frame of reference called standard position.

There are two possible directions to take. The initial side could be rotated clockwise until it arrives at the terminal side, or the initial side could be rotated counterclockwise.

In the realm of mathematics, rotations are typically counterclockwise. A counterclockwise rotation is called a positive rotation, and the corresponding angle has a positive measure. A clockwise rotation is called a negative rotation, and the corresponding angle has a negative measure.

A circle has 360 degrees. For a rotation of 360°, the terminal side and the initial side of the angle coincide.

Here are illustrations of some selected angles described in terms of rotation:

The same angle may be described as a rotation in many ways.

The angle pictured can be described as a rotation of 35° or a rotation of –325°. Furthermore, since a complete rotation has 360 degrees, the angle can also be described as a rotation of 35° + 360° = 395°, or 35° + 2(360°) = 755°, or 35° – 2(360°) = –685°, and so on.

Adding or subtracting a multiple of 360° to a rotation results in the same angle in standard position. A given angle in standard position has an associated family of infinitely many rotations, and the measures of any two rotations in the family differ by a multiple of 360°. The members of the family are known as coterminal angles because they have the same terminal side even though their rotations have different measures.

B. Reference Triangles and Trigonometric Function Values

For any angle in standard position that is not a multiple of 90°, it is possible to draw an associated right triangle called a reference triangle. The reference triangle is used to define trigonometric function values for the angle.

To determine the reference triangle, follow these steps:

Do the trigonometric function values depend on the size of the circle? What do you think?

Not surprisingly, the trigonometric function values of 30° found using the reference triangle approach do agree with the values determined in the previous section on right triangle trigonometry! However, as you will soon see, this new approach will allow to you to find trigonometric function values for angles that are not necessarily acute.

The 30° angle has its terminal side in the first quadrant. (See module 1, topic I-A for a review of the four quadrants.) The trigonometric functions of angles in other quadrants can be defined using the same process carried out for the 30° angle.

To find the trigonometric function values of an angle with terminal side in quadrants II, III, or IV, create a reference triangle and determine the ratios. Be careful to note that x or y may be negative, depending on the quadrant. The value r is always positive because it is a measure of distance. Reference triangles associated with angles in quadrants II, III, and IV are shown below.

In practice, to find the trigonometric function values of a particular angle, start with either

• the measure of the angle

or

• a particular point P on the terminal side of the angle.

In example II.B.1, you were given the angle measure of 30° and then you determined the trigonometric function values. For each of the following examples, you will be given a point P on the terminal side of the angle.

As illustrated in the preceding examples, the signs of the trigonometric function values depend on the quadrant of the angle.

Focusing on the positive values, if you remember the sequence all-sin-tan-cos, then you can quickly recall how the positive values correspond to the quadrants. This is illustrated in the circle diagram below.

Coterminal Angles

If two angles are coterminal, then their trigonometric function values are the same, because the reference triangles are the same. For example, sin 30° = sin 390° = sin(–330°).

C. Trigonometric Function Values of Multiples of 90°

For the special cases where an angle is a multiple of 90°, there is no associated reference triangle. The definitions of the trigonometric functions still apply, but some of the trigonometric values are undefined, due to denominators that are equal to 0. The following table shows how to determine the values of the sine, cosine, and tangent for the angles of 0°, 90°, 180°, and 270°. In each case, the point P chosen on the terminal side of the angle is a point on the unit circle.

D. Symmetry and Reference Angles

Recall that commonly occurring acute angles are 30°, 45°, and 60°. Commonly occurring angles between 0° and 360° include 30°, 45°, 60°, 120°, 135°, 150°, 210°, 225°, 240°, 300°, 315°, and 330°. Fortunately, if you know the trigonometric values for the acute angles, then by using symmetry, you can easily derive the exact trigonometric values for all of the angles on this list, without using a calculator!

The method used in this example relies on the idea of a reference angle.

Coterminal angles, since they have the same associated reference triangle, have the same reference angle. The calculation of the measure of the reference angle a depends upon the quadrant of the given angle, as summarized in the following table:

III. The Unit Circle and Angle Conversion

After completing this section, you should be able to:

• convert between degree and radian measures
• understand the relationship between arc length and an angle measured in radians
• find linear speed and angular speed, and apply the relationship between them

A. Radian Measure

In the discussion so far, angles have been measured in degrees. However, there is another system of measurement of angles, called radian measure, which proves to be especially convenient in scientific and mathematical settings.

The concept of a radian is related to the unit circle, the circle of radius 1 centered at the origin.

Since the circumference of a circle of radius π is 2πr, the circumference of the unit circle is 2π.

Traveling a distance π along the unit circle corresponds to the angle of π radians. Earlier, it was noted that a distance of π corresponds to the angle 180°. So, there is a simple relationship between radian measure and degree measure:

These equations can be used to convert units of measurement from degrees to radians, and from radians to degrees.

The following table lists the degree measure and the corresponding radian measure for a set of frequently used angles. The four background colors correspond to the four quadrants. The circle diagram illustrates the correspondence between degree measure and radian measure.

Considering that 180 degrees equals π radians, 360 degrees must then equal 2π radians. Recall that angles whose degree measures differ by an integer multiple of 360 degrees are coterminal. Therefore, angles whose radian measures differ by an integer multiple of 2π radians are coterminal, as well. Reference angles in radian measure are determined in an analogous fashion to reference angles in degree measure.

In the discussion so far, the focus has been placed on the unit circle. Similar ideas can be extended to any circle of radius r centered at the origin.

For angle θ, let s be the corresponding length of the arc on the circle of radius r.
Let s' be the length of the corresponding arc on the unit circle.

The ratio of the arc lengths s and s' is the same as the ratio of the radius r and 1.

That is, .

On the unit circle, the length of the arc s' is the measure of the angle θ in radians. In other words, s' = θ.

Therefore, ; equivalently, .

From this point forward, when an angle measurement is stated and no units are given, it is assumed that the measurement is in radians.

B. Linear Speed and Angular Speed

When considering applied problems, the concept of speed often arises. For example, a car might travel a linear speed of 55 miles per hour. A tire might achieve an angular speed of 200 revolutions per minute, or 400π radians per minute.

Regardless of where you sit on the disk, you will be traveling at the same angular speed, since all points on the disk will complete a 360° (or 2π radian) rotation in the same time. However, if you sit nearer the edge of the disk than the center, your linear speed will be greater. A point near the edge lies on a circle of greater radius than a point near the center. In one second, you will travel the same proportion of the circumference for either circle, but because the larger circle has a greater circumference, the distance traveled in that time will be greater, leading to a greater linear speed.

For angles measured in radians, there is a simple relationship between linear speed and angular speed. Suppose a point is located r units from the center of rotation and an arc of length s has been traversed in time t. Since s = rθ, linear speed = v = s/t = rθ/t = r, where is the angular speed, measured in radians per time unit.

To apply this relationship, be careful to make sure your units are consistent. The units of distance for v and r must be the same, the units of time for v and must be the same.

IV. Circular Functions and Their Graphs

After completing this section, you should be able to:

• find trigonometric function values for any real number
• graph the trigonometric functions and analyze their properties

A. Definitions of Circular Functions

At the beginning of this module, when we were focusing on the properties of right triangles, trigonometric function values were defined for acute angles. Next, by looking at angles in standard position, as rotations, the definitions of trigonometric functions were extended to apply to any angle. Then angles in radian measure were discussed.

This topic will extend definitions of trigonometric functions again, so that they are defined for all real numbers, rather than just for angles.

The following table summarizes trigonometric function values for certain special angles, listed in both degree and radian measure.

For every real number t, it is possible to assign corresponding values sin t and cos t.

If t is negative, apply the same ideas, but travel in the negative (clockwise) direction.

Because of the link with the unit circle, trigonometric functions of real numbers are often called circular functions.

The first three columns of the table at the beginning of this topic can be transformed to the terminology of circular functions by substituting t = θ, x = cos θ, and y = sin θ, and writing the columns in order by t, x, and y. The transformed table is shown below. The diagram to the right visually displays this information on the unit circle.

B. Graphs of the Sine and Cosine Functions

The Sine Function

To get a feel for the graph of the sine function, make a table of values of the sine function and plot the values.

y = sin t

The graph of the sine function has many interesting features.

The graph oscillates between –1 and 1. According to the definition of the sine function, any real number may be input for t, so the domain of the sine function is (–∞ , ∞). The range is [–1, 1]. The graph crosses the t-axis at each integer multiple of π, so the zeros of the sine function are 0, ±π, ±2π, ±3π, ….

Trace the graph, starting at the origin. When t = 0, the y value is 0. As t increases, the y values increase until a peak of 1 is reached when t = π/2. Then as t goes from π/2 to 3π/2, the y values decrease to –1. As t goes from 3π/2 to 2π, the y values increase to 0.

If you make a copy of the curve from 0 to 2π and place it on top of the curve from –2π to 0, the two curves coincide. The graph has a pattern that repeats.

For any value of t, sin(t + 2π) = sin t. The sine function is said to be periodic with period 2π.

Recall from college algebra that the graph of y = f(t + p) is a horizontal shift of the graph of y = f(t). (See module 1, topic III.)

If you shift the graph of y = f(t) to the left by p units, you arrive at the graph of y = f(t + p).

Saying that f(t + p) is equal to f(t) means that if you shift the graph of f to the left by p units, the graph remains the same.

In the case of the sine function, if you shift the graph of the sine function to the left by 2π, the graph is the same. If you shift the graph of the sine function to the left by less than 2π, the graph does not remain the same. The sine function is periodic with period 2π.

Alternatively, note that t and t + 2π correspond to coterminal angles, and coterminal angles have the same trigonometric function values. Therefore, sin(t + 2π) = sin t, and the sine function is periodic with p = 2π.

Given a periodic function, there are two measurements of particular interest: how often the graph repeats (the period) and how much the graph oscillates vertically (the amplitude).

In the case of the sine function, the amplitude is |1 – (–1)| = 1.

Recall that graphs may exhibit certain types of symmetry. Take a careful look at the graph of the sine function and notice that it is symmetric with respect to the origin. (See module 1, topic III.) A function f whose graph is symmetric with respect to the origin is called an odd function, and such a graph satisfies f(–t) = –f(t). Therefore, sin(–t) = –sin t.

The Cosine Function

The cosine function can be graphed by using the same approach as the sine function:

y = cos t

The graph oscillates between –1 and 1. The domain is (–∞ , ∞) and the range is [–1, 1]. The graph crosses the t-axis at , so the zeros of the cosine function are .

Trace the graph, starting when t = 0 and y = 1. As t increases, the y values decrease from the peak of 1 to –1 when t = π. Then as t goes from π to 2π, the y values increase to 1 again.

The cosine function is also periodic with period 2π and has an amplitude of 1. Notice that the cosine graph is symmetric with respect to the y-axis. A function f whose graph is symmetric with respect to the y-axis is called an even function, and satisfies f(–t) = f(t). Therefore, cos(–t) = cos t.

C. Graphs of the Other Trigonometric Functions

The Tangent Function

The domain of a function of the form consists of those inputs t in the domains of both p and q for which q(t) is nonzero.

Recalling the definitions of the circular functions, since y = sin t, x = cos t, and tan t = when x ≠ 0, the tangent function is defined as tan t =, when cos t ≠ 0.

The domain of the sine and cosine functions consists of all real numbers, and by examination of graph of the cosine function, we can see that cos t = 0 when  .

Therefore, the domain of the tangent function consists of all real numbers except .

Vertical asymptotes occur at the t values for which the denominator cos t is zero. (See module 1, topic II-F for a review of asymptotes.)

y = tan t

The tangent function has period π. The graph has infinitely many vertical asymptotes. The function has no maximum or minimum, and so the amplitude is not defined. The range is (–∞, ∞).

The graphs of the cotangent, secant, and cosecant functions can be determined using the same approach. Here is a visual comparison:

V. Transformations of Sine and Cosine Functions

After completing this section, you should be able to:

• graph transformations of sine and cosine functions

Trigonometric functions such as sine and cosine can be thought of as building blocks for other functions. The graph of a trigonometric function may be transformed by horizontal and vertical translation, reflection, vertical stretching and shrinking, and horizontal stretching and shrinking.

If you are not well acquainted with transformations of functions, it is worth reviewing module 1, topic III-B. Use the table of transformations in module 1 as a reference as you consider transformations of sine and cosine.

The ultimate goal of this topic is to graph functions of the forms

y = A sin(Bt – C) + D and y = A cos(Bt – C) + D

and to analyze their characteristics.

To get a feel for the transformations of the sine function, start by considering simple functions.

Now consider several more complicated examples which involve a sequence of transformations.

You now have all of the knowledge necessary to graph functions of the form y = A sin(Bt – C) + D.

In preparing a graph, be careful to first write the expression in the form y = A sin[B(t – C/B)] + D.

The same steps apply for graphing a function of the form y = A cos(Bt – C) + D.

Return to top of page