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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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.
Unit Circle
We defined radian measure to be
When r = 1,
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Basic Circular Functions
Reflections on a Unit Circle
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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
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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Example
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
Example
Find each of the following function values.
Solution
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Example
Example
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Example
Example
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Example
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
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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Example
Find each of the following function values of radian
measures using a calculator. Round the answers to four decimal places.
Solution:
Graph of Sine Function
[image:22.720.81.681.51.508.2]Make a table of values from the unit
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Graph of Cosine Function
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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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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.
Periodic Function
A function f is said to be periodic if there exists a positive constant p such that
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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
Amplitude of the Sine Function
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Amplitude of the Cosine Function
Odd and Even
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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.
Graph of the Tangent Function
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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
Graph of the Tangent Function
Note:
Add these ordered pairs to the graph. Use a calculator to add some other
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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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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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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.
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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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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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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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.