Download Trigonometric Functions

MAC 1114
Module 1
Trigonometric Functions
Rev.S08
Learning Objectives
•
Upon completing this module, you should be able to:
1.
2.
3.
4.
Use basic terms associated with angles.
Find measures of complementary and supplementary angles.
Calculate with degrees, minutes, and seconds.
Convert between decimal degrees and degrees, minutes, and
seconds.
Identify the characteristics of an angle in standard position.
Find measures of coterminal angles.
Find angle measures by using geometric properties.
Apply the angle sum of a triangle property.
5.
6.
7.
8.
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2
Learning Objectives (Cont.)
9.
Find angle measures and side lengths in similar triangles.
10.
Solve applications involving similar triangles.
11.
Learn basic concepts about trigonometric functions.
12.
Find function values of an angle or quadrantal angles.
13.
Decide whether a value is in the range of a trigonometric
function
14.
Use the reciprocal, Pythagorean and quotient identities.
15.
Identify the quadrant of an angle.
16.
Find other function values given one value and the quadrant.
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Trigonometric Functions
There are four major topics in this module:
- Angles
- Angle Relationships and Similar Triangles
- Trigonometric Functions
- Using the Definitions of the Trigonometric
Functions
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What are the basic terms?

Two distinct points determine a line called
line AB.
A
B

Line segment AB—a portion of the line
between A and B, including points A and B.
A

B
Ray AB—portion of line AB that starts at A
and continues through B, and on past B.
A
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B
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What are the basic terms? (cont.)

Angle-formed by rotating a
ray around its endpoint.

The ray in its initial
position is called the initial
side of the angle.

The ray in its location after
the rotation is the terminal
side of the angle.
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6
How to Identify a Positive Angle and a
Negative Angle?

Positive angle: The
rotation of the terminal
side of an angle
counterclockwise.
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
Negative angle: The
rotation of the terminal
side is clockwise.
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Most Common unit and Types of Angles


The most common unit for measuring angles is
the degree.
The major types of angles are acute angle, right
angle, obtuse angle and straight angle.
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What are Complementary Angles?

When the two angles form a right
angle, they are complementary
angles. Thus, we can find the
measure of each angle in this
case.
k +20
k  16
The two angles have measures of
43 + 20 = 63 and 43  16 = 27
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What are Supplementary Angles?

When the two angles form a straight
angle, they are supplementary
angles. Thus, we can find the measure of
each angle in this case too.
6x + 7
3x + 2
These angle measures are
6(19) + 7 = 121 and 3(19) + 2 = 59
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10
How to Convert a Degree
to Minute or Second?
Rev.S08

One minute is 1/60 of a degree.

One second is 1/60 of a minute.
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Example


Perform the calculation.

Perform the calculation.

Write
Since 86 = 60 + 26, the
sum is written
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Example

Convert
Rev.S08

Convert 36.624
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13
How to Determine an Angle is
in Standard Position?

An angle is in standard position if its vertex is
at the origin and its initial side is along the
positive x-axis.
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What are Quadrantal Angles?

Angles in standard position having their terminal
sides along the x-axis or y-axis, such as angles
with measures 90, 180, 270, and so on, are
called quadrantal angles.
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What are Coterminal Angles?

A complete rotation of a ray results in an angle
measuring 360. By continuing the rotation,
angles of measure larger than 360 can be
produced. Such angles are called coterminal
angles.
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Example




Find the angles of smallest possible positive
measure coterminal with each angle.
a) 1115
b) 187
Add or subtract 360 as may times as needed to
obtain an angle with measure greater than 0 but
less than 360.
o
o
o
a) 1115  3(360 ) = 35 b) 187 + 360 = 173
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What are Vertical Angles?

Vertical Angles have equal measures.
Q
R
M
N

P
The pair of angles NMP and RMQ are vertical
angles.
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18
Parallel Lines and Transversal


Parallel lines are lines that lie in the same
plane and do not intersect.
When a line q intersects two parallel lines, q, is
called a transversal.
Transversal
q
m
parallel lines
n
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19
Important Angle Relationships
q
m
n
Name
Angles
Rule
Alternate interior angles
4 and 5
3 and 6
Angles measures are equal.
Alternate exterior angles
1 and 8
2 and 7
Angle measures are equal.
Interior angles on the same
side of the transversal
4 and 6
3 and 5
Angle measures add to 180.
Corresponding angles
2 & 6, 1 & 5, Angle measures are equal.
3 & 7, 4 & 8
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Example of Finding Angle Measures

Find the measure of each
marked angle, given that
lines m and n are parallel.
(6x + 4)

m

(10x  80)
n


The marked angles are
alternate exterior angles,
which are equal.
Rev.S08
One angle has measure
6x + 4 = 6(21) + 4 = 130
and the other has measure
10x  80 = 10(21)  80 =
130
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21
Angle Sum of a Triangle

The sum of the measures of the angles of any
triangle is 180.
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22
Example of Applying the Angle Sum

The measures of two of
the angles of a triangle
are 52 and 65. Find the
measure of the third
angle, x.
65

Solution

The third angle of the
triangle measures 63.
x
52
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23
Types of Triangles: Angles

Note: The sum of the measures of the angles of
any triangle is 180.
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Types of Triangles: Sides

Again, the sum of the measures of the angles of
any triangle is 180.
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What are the Conditions for
Similar Triangles?

Corresponding angles must have the same
measure.

Corresponding sides must be proportional.
(That is, their ratios must be equal.)
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Example of Finding Angle Measures

Triangles ABC and DEF
are similar. Find the
measures of angles D and
E.


D
Since the triangles are
similar, corresponding
angles have the same
measure.
Angle D corresponds to
angle A which = 35
A
112
35
F
C
Rev.S08
112
33
E

Angle E corresponds to
angle B which = 33
B
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Example of Finding Side Lengths

Triangles ABC and DEF
are similar. Find the
lengths of the unknown
sides in triangle DEF.

To find side DE.

To find side FE.
D
A
16
112
35
64
F
E
32
C
112
33
B
48
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Example of Application

A lighthouse casts a
shadow 64 m long. At the
same time, the shadow
cast by a mailbox 3 feet
high is 4 m long. Find the
height of the lighthouse.

The two triangles are
similar, so corresponding
sides are in proportion.

The lighthouse is 48 m
high.
3
4
x
64
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29
The Six Trigonometric Functions

Let (x, y) be a point other the origin on the terminal
side of an angle  in standard position. The
distance from the point to the origin is
The six trigonometric functions of  are defined as
follows.
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Example of Finding Function Values

The terminal side of angle  in standard position
passes through the point (12, 16). Find the
values of the six trigonometric functions of
angle  .
(12, 16)
16

12
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31
Example of Finding Function Values (cont.)

Since x = 12, y = 16, and r = 20, we have
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Another Example

Find the six trigonometric
function values of the
angle  in standard
position, if the terminal
side of  is defined by
x + 2y = 0, x  0.

We can use any point on
the terminal side of  to
find the trigonometric
function values.
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Another Example (cont.)


Choose x = 2

Use the definitions:
The point (2, 1) lies on
the terminal side, and the
corresponding value of r is
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Example of Finding Function Values with
Quadrantal Angles


Find the values of the six trigonometric functions for an angle
of 270.
First, we select any point on the terminal side of a 270 angle.
We choose (0, 1). Here x = 0, y = 1 and r = 1.
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Undefined Function Values

If the terminal side of a quadrantal angle lies
along the y-axis, then the tangent and secant
functions are undefined.

If it lies along the x-axis, then the cotangent
and cosecant functions are undefined.
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What are the Commonly Used
Function Values?
sin 
cos 
tan 
cot 
sec 
csc 
0
0
1
0
undefined
1
undefined
90
1
0
undefined
0
undefined
1
180
0
1
0
undefined
1
undefined
270
1
0
undefined
0
undefined
1
360
0
1
0
undefined
1
undefined

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Reciprocal Identities
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38
Example of Finding Function Values
Using Reciprocal Identities


Find cos  if sec  =

Find sin  if csc 
Since cos  is the
reciprocal of sec 
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Signs of Function Values
at Different Quadrants
 in
sin 
Quadrant
cos 
tan 
cot 
sec 
csc 
I
+
+
+
+
+
+
II
+




+
III


+
+


IV

+


+

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Identify the Quadrant




Identify the quadrant (or quadrants) of any
angle  that satisfies tan  > 0 and cot  > 0.
tan  > 0 in quadrants I and III
cot  > 0 in quadrants I and III
so, the answer is quadrants I and III
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41
Ranges of Trigonometric Functions




For any angle  for which the indicated functions
exist:
1. 1  sin   1 and 1  cos   1;
2. tan  and cot  can equal any real number;
3. sec   1 or sec   1 and
csc   1 or csc   1.
(Notice that sec  and csc  are never between
1 and 1.)
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Pythagorean Identities
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43
Quotient Identities
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Example of Other Function Values



Find sin  and cos  if tan  = 4/3 and  is in
quadrant III.
Since  is in quadrant III, sin  and cos  will
both be negative.
sin  and cos  must be in the interval [1, 1].
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Example of Other Function Values (cont.)

We use the identity
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What have we learned?
•
We have learned to
1.
2.
3.
4.
Use basic terms associated with angles.
Find measures of complementary and supplementary angles.
Calculate with degrees, minutes, and seconds.
Convert between decimal degrees and degrees, minutes, and
seconds.
Identify the characteristics of an angle in standard position.
Find measures of coterminal angles.
Find angle measures by using geometric properties.
Apply the angle sum of a triangle property.
5.
6.
7.
8.
Rev.S08
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47
What have we learned? (Cont.)
9.
Find angle measures and side lengths in similar triangles.
10.
Solve applications involving similar triangles.
11.
Learn basic concepts about trigonometric functions.
12.
Find function values of an angle or quadrantal angles.
13.
Decide whether a value is in the range of a trigonometric
function
14.
Use the reciprocal, Pythagorean and quotient identities.
15.
Identify the quadrant of an angle.
16.
Find other function values given one value and the quadrant.
Rev.S08
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48
Credit
•
Some of these slides have been adapted/modified in part/whole from
the slides of the following textbook:
•
Margaret L. Lial, John Hornsby, David I. Schneider, Trigonometry, 8th
Edition
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49