1.3 Trigonometric Functions

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.

Radian Measure

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 $θ,$ let $s$ be the length of the corresponding arc on the unit circle (Figure 1.30). We say the angle corresponding to the arc of length 1 has radian measure 1.

An image of a circle. At the exact center of the circle there is a point. From this point, there is one line segment that extends horizontally to the right a point on the edge of the circle and another line segment that extends diagonally upwards and to the right to another point on the edge of the circle. These line segments have a length of 1 unit. The curved segment on the edge of the circle that connects the two points at the end of the line segments is labeled “s”. Inside the circle, there is an arrow that points from the horizontal line segment to the diagonal line segment. This arrow has the label “theta = s radians”. — Figure 1.30The radian measure of an angle $θ$ is the arc length $s$ of the associated arc on the unit circle.

Since an angle of $360 °$ corresponds to the circumference of a circle, or an arc of length $2 π,$ we conclude that an angle with a degree measure of $360 °$ has a radian measure of $2 π .$ Similarly, we see that $180 °$ is equivalent to $π$ radians. Table 1.8 shows the relationship between common degree and radian values.

Table 1.8Common Angles Expressed in Degrees and Radians

Degrees	Radians	Degrees	Radians
0	0	120	$2 π / 3$
30	$π / 6$	135	$3 π / 4$
45	$π / 4$

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 $P = (x, y)$ on the unit circle. Let $θ$ be an angle with an initial side that lies along the positive $x$ -axis and with a terminal side that is the line segment $O P .$ An angle in this position is said to be in standard position (Figure 1.31). We can then define the values of the six trigonometric functions for $θ$ in terms of the coordinates $x$ and $y .$

An image of a graph. The graph has a circle plotted on it, with the center of the circle at the origin, where there is a point. From this point, there is one line segment that extends horizontally along the x axis to the right to a point on the edge of the circle. There is another line segment that extends diagonally upwards and to the right to another point on the edge of the circle. This point is labeled “P = (x, y)”. These line segments have a length of 1 unit. From the point “P”, there is a dotted vertical line that extends downwards until it hits the x axis and thus the horizontal line segment. Inside the circle, there is an arrow that points from the horizontal line segment to the diagonal line segment. This arrow has the label “theta”. — Figure 1.31The angle $θ$ is in standard position. The values of the trigonometric functions for $θ$ are defined in terms of the coordinates $x$ and $y .$

We can see that for a point $P = (x, y)$ on a circle of radius $r$ with a corresponding angle $θ,$ the coordinates $x$ and $y$ satisfy

cos θ = \frac{x}{r} x = r cos θ

sin θ = \frac{y}{r} y = r sin θ .

The values of the other trigonometric functions can be expressed in terms of $x, y,$ and $r$ (Figure 1.32).

Table 1.9 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 $sin θ$ and $cos θ .$

Table 1.9Values of

sin θ

and

cos θ

at Major Angles

θ

in the First Quadrant

$θ$	$s in θ$	$cos θ$
$0$	$0$

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 $θ$ be one of the acute angles. Let $A$ be the length of the adjacent leg, $O$ be the length of the opposite leg, and $H$ be the length of the hypotenuse. By inscribing the triangle into a circle of radius $H,$ as shown in Figure 1.33, we see that $A, H,$ and $O$ satisfy the following relationships with $θ :$

sin θ = \frac{O}{H} cos θ = \frac{A}{H} tan θ = \frac{O}{A} csc θ = \frac{H}{O} sec θ = \frac{H}{A} cot θ = \frac{A}{O}

Trigonometric Identities

A trigonometric identity is an equation involving trigonometric functions that is true for all angles $θ$ 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.

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 $P = (x, y)$ be a point on the unit circle and let $θ$ be the corresponding angle $.$ Since the angle $θ$ and $θ + 2 π$ correspond to the same point $P,$ the values of the trigonometric functions at $θ$ and at are the same. Consequently, the trigonometric functions are 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 1.34).

An image of six graphs. Each graph has an x axis that runs from -2 pi to 2 pi and a y axis that runs from -2 to 2. The first graph is of the function “f(x) = sin(x)”, which is a curved wave function. The graph of the function starts at the point (-2 pi, 0) and increases until the point (-((3 pi)/2), 1). After this point, the function decreases until the point (-(pi/2), -1). After this point, the function increases until the point ((pi/2), 1). After this point, the function decreases until the point (((3 pi)/2), -1). After this point, the function begins to increase again. The x intercepts shown on the graph are at the points (-2 pi, 0), (-pi, 0), (0, 0), (pi, 0), and (2 pi, 0). The y intercept is at the origin. The second graph is of the function “f(x) = cos(x)”, which is a curved wave function. The graph of the function starts at the point (-2 pi, 1) and decreases until the point (-pi, -1). After this point, the function increases until the point (0, 1). After this point, the function decreases until the point (pi, -1). After this point, the function increases again. The x intercepts shown on the graph are at the points (-((3 pi)/2), 0), (-(pi/2), 0), ((pi/2), 0), and (((3 pi)/2), 0). The y intercept is at the point (0, 1). The graph of cos(x) is the same as the graph of sin(x), except it is shifted to the left by a distance of (pi/2). On the next four graphs there are dotted vertical lines which are not a part of the function, but act as boundaries for the function, boundaries the function will never touch. They are known as vertical asymptotes. There are infinite vertical asymptotes for all of these functions, but these graphs only show a few. The third graph is of the function “f(x) = csc(x)”. The vertical asymptotes for “f(x) = csc(x)” on this graph occur at “x = -2 pi”, “x = -pi”, “x = 0”, “x = pi”, and “x = 2 pi”. Between the “x = -2 pi” and “x = -pi” asymptotes, the function looks like an upward facing “U”, with a minimum at the point (-((3 pi)/2), 1). Between the “x = -pi” and “x = 0” asymptotes, the function looks like an downward facing “U”, with a maximum at the point (-(pi/2), -1). Between the “x = 0” and “x = pi” asymptotes, the function looks like an upward facing “U”, with a minimum at the point ((pi/2), 1). Between the “x = pi” and “x = 2 pi” asymptotes, the function looks like an downward facing “U”, with a maximum at the point (((3 pi)/2), -1). The fourth graph is of the function “f(x) = sec(x)”. The vertical asymptotes for this function on this graph are at “x = -((3 pi)/2)”, “x = -(pi/2)”, “x = (pi/2)”, and “x = ((3 pi)/2)”. Between the “x = -((3 pi)/2)” and “x = -(pi/2)” asymptotes, the function looks like an downward facing “U”, with a maximum at the point (-pi, -1). Between the “x = -(pi/2)” and “x = (pi/2)” asymptotes, the function looks like an upward facing “U”, with a minimum at the point (0, 1). Between the “x = (pi/2)” and “x = (3pi/2)” asymptotes, the function looks like an downward facing “U”, with a maximum at the point (pi, -1). The graph of sec(x) is the same as the graph of csc(x), except it is shifted to the left by a distance of (pi/2). The fifth graph is of the function “f(x) = tan(x)”. The vertical asymptotes of this function on this graph occur at “x = -((3 pi)/2)”, “x = -(pi/2)”, “x = (pi/2)”, and “x = ((3 pi)/2)”. In between all of the vertical asymptotes, the function is always increasing but it never touches the asymptotes. The x intercepts on this graph occur at the points (-2 pi, 0), (-pi, 0), (0, 0), (pi, 0), and (2 pi, 0). The y intercept is at the origin. The sixth graph is of the function “f(x) = cot(x)”. The vertical asymptotes of this function on this graph occur at “x = -2 pi”, “x = -pi”, “x = 0”, “x = pi”, and “x = 2 pi”. In between all of the vertical asymptotes, the function is always decreasing but it never touches the asymptotes. The x intercepts on this graph occur at the points (-((3 pi)/2), 0), (-(pi/2), 0), ((pi/2), 0), and (((3 pi)/2), 0) and there is no y intercept.

Just as with algebraic functions, we can apply transformations to trigonometric functions. In particular, consider the following function:

f (x) = A cos (B (x - α)) + C .

In Figure 1.35, the constant $α$ causes a horizontal or phase shift. The factor $B$ changes the period. This transformed sine function will have a period $2 π / ∣ B ∣.$ The factor $A$ results in a vertical stretch by a factor of $∣ A ∣.$ We say $∣ A ∣$ is the “amplitude of $f .$ ” The constant $C$ causes a vertical shift.

An image of a graph. The graph is of the function “f(x) = Acos(B(x - alpha)) + C”. Along the y axis, there are 3 hash marks: starting from the bottom and moving up, the hash marks are at the values “C - A”, “C”, and “C + A”. The distance from the origin to “C” is labeled “vertical shift”. The distance from “C - A” to “A” and the distance from “A” to “C + A” is “A”, which is labeled “amplitude”. On the x axis is a hash mark at the value “alpha” and the distance between the origin and “alpha” is labeled “horizontal shift”. The distance between two successive minimum values of the function (in other words, the distance between two bottom parts of the wave that are next to each other) is “(2 pi)/(absolute value of B)” is labeled the period. The period is also the distance between two successive maximum values of the function. — Figure 1.35A graph of a general cosine function.

Notice in Figure 1.34 that the graph of $y = cos x$ is the graph of $y = sin x$ shifted to the left $π / 2$ units. Therefore, we can write $cos x = sin (x + π / 2) .$ Similarly, we can view the graph of $y = sin x$ 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

h (t) = 3.7 sin (\frac{2 π}{365} (t - 80.5)) + 12

is a model for the number of hours of daylight $h$ as a function of day of the year $t$ (Figure 1.36).

An image of a graph. The x axis runs from 0 to 365 and is labeled “t, day of the year”. The y axis runs from 0 to 20 and is labeled “h, number of daylight hours”. The graph is of the function “h(t) = 3.7sin(((2 pi)/365)(t - 80.5)) + 12”, which is a curved wave function. The function starts at the approximate point (0, 8.4) and begins increasing until the approximate point (171.8, 15.7). After this point, the function decreases until the approximate point (354.3, 8.3). After this point, the function begins increasing again. — Figure 1.36The hours of daylight as a function of day of the year can be modeled by a shifted sine curve.

Key Concepts

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 $π$ rad.
For acute angles $θ,$ 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 $θ .$
For a general angle $θ,$ let $(x, y)$ be a point on a circle of radius $r$ corresponding to this angle $θ .$ The trigonometric functions can be written as ratios involving and

Key Equations

Generalized sine function

f (x) = A sin (B (x - α)) + C