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CHAPTER 6: The Trigonometric Functions

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CHAPTER 6:

The Trigonometric Functions

6.1The Trigonometric Functions of Acute Angles 6.2Applications of Right Triangles

6.3Trigonometric Functions of Any Angle 6.4Radians, Arc Length, and Angular Speed

6.5Circular Functions: Graphs and Properties

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Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

6.5

Circular Functions: Graphs and Properties

∙ 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.

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Unit Circle

We defined radian measure to be

When r = 1,

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Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Basic Circular Functions

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Reflections on a Unit Circle

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Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

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 the

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Example

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

origin.

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Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Example

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Example

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Find Function Values

Knowing only a few points on the unit circle allows us to find

trigonometric function

values of frequently

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Example

Find each of the following function values.

Solution

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Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Example

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Example

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Example

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Example

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Example

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Example

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Signs of Other Trig Functions by Quadrant

of Angle

■ Reciprocal functions will always have the same

sign

■ All functions have positive values for angles in

Quadrant I

■ Sine and Cosecant have positive values for

angles in Quadrant II

■ Tangent and Cotangent have positive values for

angles in Quadrant III

■ Cosine and Secant have positive values for

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Memorizing Signs of Trig Functions by

Quadrant

■ It will help to memorize by learning these words in

Quadrants I - IV:

“All students take calculus”

And remembering reciprocal identities

■ Trig functions are negative in quadrants where they are

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Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Example

Find each of the following function values of radian

measures using a calculator. Round the answers to four decimal places.

Solution:

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Graph of Sine Function

[image:22.720.81.681.51.508.2]

Make a table of values from the unit

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Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Graph of Cosine Function

[image:25.720.62.678.53.508.2]
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Domain and Range of Sine and Cosine

Functions

The domain of the sine function and the cosine function is (–∞, ∞).

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Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

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.

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Periodic Function

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

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Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

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

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Amplitude of the Sine Function

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Amplitude of the Cosine Function

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Odd and Even

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Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Odd and Even

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

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

The sine function is odd.

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Graph of the Tangent Function

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Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Graph of the Tangent Function

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

Draw vertical

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Graph of the Tangent Function

Note:

Add these ordered pairs to the graph. Use a calculator to add some other

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Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

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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.

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Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

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, …

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Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

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.

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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

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Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

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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.

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Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

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

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Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

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.

Figure

table of values from the
table of values from the

References

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