1. Functions and Graphs
- Convert angle measures between degrees and radians.
- Recognize the triangular and circular definitions of the basic trigonometric functions.
- Write the basic trigonometric identities.
- Identify the graphs and periods of the trigonometric functions.
- Describe the shift of a sine or cosine graph from the equation of the function.
Trigonometric functions are used to model many phenomena, including sound waves, vibrations of strings, alternating electrical current, and the motion of pendulums. In fact, almost any repetitive, or cyclical, motion can be modeled by some combination of trigonometric functions. In this section, we define the six basic trigonometric functions and look at some of the main identities involving these functions.
To use trigonometric functions, we first must understand how to measure the angles. Although we can use both radians and degrees, radians are a more natural measurement because they are related directly to the unit circle, a circle with radius 1. The radian measure of an angle is defined as follows. Given an angle , let be the length of the corresponding arc on the unit circle ((Figure)). We say the angle corresponding to the arc of length 1 has radian measure 1.
Since an angle of 360° corresponds to the circumference of a circle, or an arc of length , we conclude that an angle with a degree measure of 360° has a radian measure of . Similarly, we see that 180° is equivalent to radians. (Figure) shows the relationship between common degree and radian values.
Converting between Radians and Degrees
- Express 225° using radians.
- Express rad using degrees.
Use the fact that 180° is equivalent to radians as a conversion factor: .
- rad =
Express 210° using radians. Express rad using degrees.
The Six Basic Trigonometric Functions
Trigonometric functions allow us to use angle measures, in radians or degrees, to find the coordinates of a point on any circle—not only on a unit circle—or to find an angle given a point on a circle. They also define the relationship among the sides and angles of a triangle.
To define the trigonometric functions, first consider the unit circle centered at the origin and a point on the unit circle. Let be an angle with an initial side that lies along the positive -axis and with a terminal side that is the line segment . An angle in this position is said to be in standard position ((Figure)). We can then define the values of the six trigonometric functions for in terms of the coordinates and .
Let be a point on the unit circle centered at the origin . Let be an angle with an initial side along the positive -axis and a terminal side given by the line segment . The trigonometric functions are then defined as
If , then and are undefined. If , then and are undefined.
We can see that for a point on a circle of radius with a corresponding angle , the coordinates and satisfy
The values of the other trigonometric functions can be expressed in terms of , and ((Figure)).
(Figure) shows the values of sine and cosine at the major angles in the first quadrant. From this table, we can determine the values of sine and cosine at the corresponding angles in the other quadrants. The values of the other trigonometric functions are calculated easily from the values of and .
Evaluating Trigonometric Functions
Evaluate each of the following expressions.
- On the unit circle, the angle corresponds to the point . Therefore, .
- An angle corresponds to a revolution in the negative direction, as shown. Therefore, .
- An angle . Therefore, this angle corresponds to more than one revolution, as shown. Knowing the fact that an angle of corresponds to the point , we can conclude that .
Evaluate and .
As mentioned earlier, the ratios of the side lengths of a right triangle can be expressed in terms of the trigonometric functions evaluated at either of the acute angles of the triangle. Let be one of the acute angles. Let be the length of the adjacent leg, be the length of the opposite leg, and be the length of the hypotenuse. By inscribing the triangle into a circle of radius , as shown in (Figure), we see that , and satisfy the following relationships with :
Constructing a Wooden Ramp
A wooden ramp is to be built with one end on the ground and the other end at the top of a short staircase. If the top of the staircase is 4 ft from the ground and the angle between the ground and the ramp is to be , how long does the ramp need to be?
Let denote the length of the ramp. In the following image, we see that needs to satisfy the equation . Solving this equation for , we see that ft.
A house painter wants to lean a 20-ft ladder against a house. If the angle between the base of the ladder and the ground is to be , how far from the house should she place the base of the ladder?
A trigonometric identity is an equation involving trigonometric functions that is true for all angles for which the functions are defined. We can use the identities to help us solve or simplify equations. The main trigonometric identities are listed next.
Rule: Trigonometric Identities
Addition and subtraction formulas
Solving Trigonometric Equations
For each of the following equations, use a trigonometric identity to find all solutions.
[reveal-answer q=”501288″]Show Answer[/reveal-answer][hidden-answer a=”501288″]a. Using the double-angle formula for , we see that is a solution of
if and only if
which is true if and only if
To solve this equation, it is important to note that we need to factor the left-hand side and not divide both sides of the equation by . The problem with dividing by is that it is possible that is zero. In fact, if we did divide both sides of the equation by , we would miss some of the solutions of the original equation. Factoring the left-hand side of the equation, we see that is a solution of this equation if and only if
, or ,
we conclude that the set of solutions to this equation is
, and .
b. Using the double-angle formula for and the reciprocal identity for , the equation can be written as
To solve this equation, we multiply both sides by to eliminate the denominator, and say that if satisfies this equation, then satisfies the equation
However, we need to be a little careful here. Even if satisfies this new equation, it may not satisfy the original equation because, to satisfy the original equation, we would need to be able to divide both sides of the equation by . However, if , we cannot divide both sides of the equation by . Therefore, it is possible that we may arrive at extraneous solutions. So, at the end, it is important to check for extraneous solutions. Returning to the equation, it is important that we factor out of both terms on the left-hand side instead of dividing both sides of the equation by . Factoring the left-hand side of the equation, we can rewrite this equation as
Therefore, the solutions are given by the angles such that or . The solutions of the first equation are . The solutions of the second equation are . After checking for extraneous solutions, the set of solutions to the equation is
Find all solutions to the equation .
Proving a Trigonometric Identity
Prove the trigonometric identity .
We start with the identity
Dividing both sides of this equation by , we obtain
Since and , we conclude that
Prove the trigonometric identity .
Graphs and Periods of the Trigonometric Functions
We have seen that as we travel around the unit circle, the values of the trigonometric functions repeat. We can see this pattern in the graphs of the functions. Let be a point on the unit circle and let be the corresponding angle. Since the angle and correspond to the same point , the values of the trigonometric functions at and at are the same. Consequently, the trigonometric functions are periodic functions. The period of a function is defined to be the smallest positive value such that for all values in the domain of . The sine, cosine, secant, and cosecant functions have a period of . Since the tangent and cotangent functions repeat on an interval of length , their period is ((Figure)).
Just as with algebraic functions, we can apply transformations to trigonometric functions. In particular, consider the following function:
In (Figure), the constant causes a horizontal or phase shift. The factor changes the period. This transformed sine function will have a period . The factor results in a vertical stretch by a factor of . We say is the “amplitude of .” The constant causes a vertical shift.
Notice in (Figure) that the graph of is the graph of shifted to the left units. Therefore, we can write . Similarly, we can view the graph of as the graph of shifted right units, and state that .
A shifted sine curve arises naturally when graphing the number of hours of daylight in a given location as a function of the day of the year. For example, suppose a city reports that June 21 is the longest day of the year with 15.7 hours and December 21 is the shortest day of the year with 8.3 hours. It can be shown that the function
is a model for the number of hours of daylight as a function of day of the year ((Figure)).
Sketching the Graph of a Transformed Sine Curve
Sketch a graph of .
This graph is a phase shift of to the right by units, followed by a horizontal compression by a factor of 2, a vertical stretch by a factor of 3, and then a vertical shift by 1 unit. The period of is .
Describe the relationship between the graph of and the graph of .
To graph , the graph of needs to be compressed horizontally by a factor of 4, then stretched vertically by a factor of 3, then shifted down 5 units. The function will have a period of and an amplitude of 3.
- Radian measure is defined such that the angle associated with the arc of length 1 on the unit circle has radian measure 1. An angle with a degree measure of 180° has a radian measure of rad.
- For acute angles , the values of the trigonometric functions are defined as ratios of two sides of a right triangle in which one of the acute angles is .
- For a general angle , let be a point on a circle of radius corresponding to this angle . The trigonometric functions can be written as ratios involving , and .
- The trigonometric functions are periodic. The sine, cosine, secant, and cosecant functions have period . The tangent and cotangent functions have period .
- Generalized sine function
For the following exercises, convert each angle in degrees to radians. Write the answer as a multiple of .
For the following exercises, convert each angle in radians to degrees.
Evaluate the following functional values.
For the following exercises, consider triangle , a right triangle with a right angle at . a. Find the missing side of the triangle. b. Find the six trigonometric function values for the angle at . Where necessary, round to one decimal place.
For the following exercises, is a point on the unit circle. a. Find the (exact) missing coordinate value of each point and b. find the values of the six trigonometric functions for the angle with a terminal side that passes through point . Rationalize all denominators.
For the following exercises, simplify each expression by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only.
For the following exercises, verify that each equation is an identity.
For the following exercises, solve the trigonometric equations on the interval .
For the following exercises, each graph is of the form or , where . Write the equation of the graph.
For the following exercises, find a. the amplitude, b. the period, and c. the phase shift with direction for each function.
a. 1 b. c. units to the right
a. b. c. No phase shift
a. 3 b. 2 c. units to the left
61. [T] The diameter of a wheel rolling on the ground is 40 in. If the wheel rotates through an angle of , how many inches does it move? Approximate to the nearest whole inch.
Approximately 42 in.
62. [T] Find the length of the arc intercepted by central angle in a circle of radius . Round to the nearest hundredth.
a. cm, rad
b. cm, rad
63. [T] As a point moves around a circle, the measure of the angle changes. The measure of how fast the angle is changing is called angular speed, , and is given by , where is in radians and is time. Find the angular speed for the given data. Round to the nearest thousandth.
a. rad, sec
b. rad, sec
c. rad, min
d. rad, min
a. 0.550 rad/sec b. 0.236 rad/sec c. 0.698 rad/min d. 1.697 rad/min
64. [T] A total of 250,000 m2 of land is needed to build a nuclear power plant. Suppose it is decided that the area on which the power plant is to be built should be circular.
- Find the radius of the circular land area.
- If the land area is to form a 45° sector of a circle instead of a whole circle, find the length of the curved side.
65. [T] The area of an isosceles triangle with equal sides of length is
where is the angle formed by the two sides. Find the area of an isosceles triangle with equal sides of length 8 in. and angle rad.
66. [T] A particle travels in a circular path at a constant angular speed . The angular speed is modeled by the function . Determine the angular speed at sec.
67. [T] An alternating current for outlets in a home has voltage given by the function
where is the voltage in volts at time in seconds.
- Find the period of the function and interpret its meaning.
- Determine the number of periods that occur when 1 sec has passed.
a. ; the voltage repeats every sec b. Approximately 59 periods
68. [T] The number of hours of daylight in a northeast city is modeled by the function
where is the number of days after January 1.
- Find the amplitude and period.
- Determine the number of hours of daylight on the longest day of the year.
- Determine the number of hours of daylight on the shortest day of the year.
- Determine the number of hours of daylight 90 days after January 1.
- Sketch the graph of the function for one period starting on January 1.
69. [T] Suppose that is a mathematical model of the temperature (in degrees Fahrenheit) at hours after midnight on a certain day of the week.
- Determine the amplitude and period.
- Find the temperature 7 hours after midnight.
- At what time does ?
- Sketch the graph of over .
a. Amplitude = 10; period = 24 b. c. 14 hours later, or 2 p.m. d.
70. [T] The function models the height (in feet) of the tide hours after midnight. Assume that is midnight.
- Find the amplitude and period.
- Graph the function over one period.
- What is the height of the tide at 4:30 a.m.?
- periodic function
- a function is periodic if it has a repeating pattern as the values of move from left to right
- for a circular arc of length on a circle of radius 1, the radian measure of the associated angle is
- trigonometric functions
- functions of an angle defined as ratios of the lengths of the sides of a right triangle
- trigonometric identity
- an equation involving trigonometric functions that is true for all angles for which the functions in the equation are defined