Section 6.5

Circular Functions: Graphs and Properties

Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.

Objectives

Given the coordinates of a point on the unit circle, find its reflections across the x-axis, the y-axis, and the origin.

Determine the six trigonometric function values for a real number when the coordinates of the point on the unit circle determined by that real number are given.

Find the function values for any real number using a calculator.

Graph the six circular functions and state their properties.

Unit Circle

We defined radian measure to be

When r = 1,

s

r

s

1, or s

The arc length s on a unit circle is the same as the radian measure of the angle .

Basic Circular Functions

For a real number s that determines a point (x, y) on the unit circle:sin s second coordinate y cos s first coordinate x

tan s second coordinate

first coordinatey

xx 0

csc s 1

second coordinate

1

yy 0

sec s 1

first coordinate

1

xx 0

cot s first coordinate

second coordinatex

yy 0

Reflections on a Unit Circle

Let’s consider the radian measure π/3 and determine the coordinates of the point on the unit circle.

Reflections on a Unit Circle

We have a 30º- 60º right triangle with hypotenuse 1 and side opposite 30º 1/2 the hypotenuse, or 1/2. This is the x-coordinate of the point. Let’s find they-coordinate.

1

2

2

y2 1

y2 1 1

4

3

4

y 3

4

3

2

Example

Each of the following points lies on the unit circle. Find their reflections across the x-axis, the y-axis, and the origin.

Solution:

a) 3

5,4

5

b)

2

2,

2

2

c)

1

2,

3

2

a)

Example (cont)

b) 2

2,

2

2

Example (cont)

c) 1

2,

3

2

Find Function Values

Knowing only a few points on the unit circle allows us to find trigonometric function values of frequently used numbers.

Example

Find each of the following function values.

a) tan3

b) cos34

c) sin 6

d) cos

43

e) cot f) csc 72

Solution

Locate the point on the unit circle determined by the rotation, and then find its coordinates using reflection if necessary.

Example(cont)

Solution continued

a) tan3

tan3

y

x

3 2

1 2

3

Example (cont)

b) cos34

cos34

x

2

2

Example (cont)

c) sin 6

sin 6

y

1

2

Example (cont)

d) cos43

cos43

x

1

2

Example (cont)

e) cot

cot x

y

1

0

which is not defined

Example (cont)

f) csc 72

csc 72

1

y

1

1

1

Example

Find each of the following function values of radian measures using a calculator. Round the answers to four decimal places.

Solution:

With the calculator in RADIAN mode:

a) cos25

b) tan 3 c) sin24.9 d) sec7

a) cos25

0.3090 b) tan 3 0.1425

c) sin24.9 0.2306d) sec

7

1

cos7

1.1099

Graph of Sine Function

Make a table of values from the unit circle.

Graph of Sine Function

Graph of Cosine Function

Make a table of values from the unit circle.

Graph of Cosine Function

Domain and Range of Sine and Cosine FunctionsThe domain of the sine function and the cosine function is (–∞, ∞).

The range of the sine function and the cosine function is [–1, 1].

Periodic Function

A function with a repeating pattern is called periodic. The sine and cosine functions are periodic because they repeat themselves every 2π units.

To see this another way, think of the part of the graph between 0 and 2π and note that the rest of the graph consists of copies of it.

The sine and cosine functions each have a period of 2π.

The period can be thought of as the length of the shortest recurring interval.

Periodic Function

A function f is said to be periodic if there exists a positive constant p such that

for all s in the domain of f. The smallest such positive number p is called the period of the function.

f s p f s

sin s 2 sin s cos s 2 cos s

Amplitude

The amplitude of a periodic function is defined to be one half the distance between its maximum and minimum function values. It is always positive.

Both the graphs and the unit circle verify that the maximum value of the sine and cosine functions is 1, whereas the minimum value of each is –1.

Amplitude of the Sine Function

the amplitude of the sine function 1

21 1 1

Amplitude of the Cosine Function

the amplitude of the cosine function 1

21 1 1

Odd and Even

Consider any real number s and its opposite, –s. These numbers determine points T and T1.

Odd and Even

Because their second coordinates are opposites of each other, we know that for any number s,

sin s sin s

Because their first coordinates are opposites of each other, we know that for any number s,

cos s cos s

The sine function is odd.

The cosine function is even.

Graph of the Tangent Function

Instead of a table, let’s begin with the definition and a few points on the unit circle.

sintan

cos

y ss

x s

Graph of the Tangent Function

Tangent function is not defined when x, the first coordinate, is 0; that is, when cos s = 0:

s 2

, 32

, 52

, ...

Draw vertical asymptotes at these locations.

Graph of the Tangent Function

Note:tan s 0 at s 0, , 2 , 3 ,...

Add these ordered pairs to the graph. Use a calculator to add some other points in (–π/2, π/2).

tan s 1 at s ... 74

, 34

,4

,54

,94

...

tan s 1 at s ... 94

, 54

, 4

,34

,74

...

Graph of the Tangent Function

Now we can complete the graph.

Graph of the Tangent Function

From the graph, we see that:

Period is π.

There is no amplitude (no maximum or minimum values).

Domain is the set of all real numbers except (π/2) + kπ, where k is an integer.

Range is the set of all real numbers.

Graph of the Cotangent Function

The cotangent function (cot s = cos s/sin s) is not defined when y, the second coordinate, is 0; that is, it is not defined for any number s whose sine is 0.

Cotangent is not defined for s = 0, ±2π, ±3π, …

The graph of the cotangent function is on the next slide.

Graph of the Cotangent Function

Graph of the Cotangent Function

From the graph, we see that:

Period is π.

There is no amplitude (no maximum or minimum values).

Domain is the set of all real numbers except kπ, where k is an integer.

Range is the set of all real numbers.

Graph of the Cosecant Function

The cosecant and sine functions are reciprocals.

The graph of the cosecant function can be constructed by finding the reciprocals of the values of the sine function. The cosecant function is not defined for those values of s whose sine is 0.

The graph of the cosecant function is on the next slide with the graph of the sine function in gray for reference.

Graph of the Cosecant Function

Graph of the Cosecant Function

From the graph, we see that:

Period is 2π.

There is no amplitude (no maximum or minimum values).

Domain is the set of all real numbers except kπ, where k is an integer.

Range is (–∞, –1] U [1, ∞).

Graph of the Secant Function

The secant and cosine functions are reciprocals.The graph of the secant function can be constructed by finding the reciprocals of the values of the cosine function.

The secant function is not defined for those values of s whose cosine is 0.

The graph of the secant function is on the next slide with the graph of the cosine function in gray for reference.

Graph of the Cosecant Function

Graph of the Secant Function

From the graph, we see that:

Period is 2π.

There is no amplitude (no maximum or minimum values).

Domain is the set of all real numbers except kπ, where k is an integer.

Range is (–∞, –1] U [1, ∞).