Chapter 4 Review of the Trigonometric Functions

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Chapter 4 Review of the Trigonometric Functions
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Standard Position
Vertex at origin
The initial side of an angle in standard
position is always located on the positive
x-axis.
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Positive and negative angles
When sketching angles, always use an arrow to
show direction.
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Classifying Angles
Angles in standard position are often classified
according to the quadrant in which their terminal
sides lie. Example 50º is a 1st quadrant
angle. 208º is a 3rd quadrant angle.
II I -75º is a 4th quadrant angle.
III IV
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Classifying Angles
Standard position angles that have their terminal
side on one of the axes are called ______________
angles. For example, 0º, 90º, 180º, 270º, 360º,
are quadrantal angles.
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Degrees, minutes, and seconds

• 1 minute (1') degree () OR 1
  ______ '
• 1 second (1") _____ minute (') OR 1'
  _______"
• Therefore, 1 second (1") ________ degree ()
• Example
• Convert to decimal degrees (to three decimal
  places)

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Degrees, minutes, and seconds

• Conversions between decimal degrees and degrees,
  minutes, seconds can be easily accomplished using
  your TI graphing calculator.
• The ANGLE menu ???on your calculator has built-in
  features for converting between decimal degrees
  and DMS.

Note that the seconds (?) symbol is not in the
ANGLE menu. Use ?? for ? symbol.
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Practice
NOTE SET MODE TO DEGREE

• Using your TI graphing calculator,
• 1) Convert to decimal degrees
  to the nearest hundredth of a degree.
• 2) Convert 57.328 to an equivalent angle
  expressed to the nearest second.

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Coterminal Angles
Angles that have the same initial and terminal
sides are coterminal.
Angles ? and ? are coterminal.
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Examples of Coterminal Angles

• Find one positive and one negative coterminal
  angle for each angle given.
• a) 125? b) 240? 34' c) ?311.8?

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The sides of a right triangle
Take a look at the right triangle, with an acute
angle, ?, in the figure below. Notice how the
three sides are labeled in reference to ?.

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Definitions of the Six Trigonometric Functions
To remember the definitions of Sine, Cosine and
Tangent, we use the acronym SOH CAH
TOA
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Definitions of the Trig Functions
Definitions of Trigonometric Functions of an
Angle Let ? be an angle in standard position with
(x, y) a point on the terminal side of ? and r
is the distance from the origin to the point.
Using the Pythagorean theorem, we have
.
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Example
Let (12, 5) be a point on the terminal side of ?.
Find the value of the six trig functions of ?.

First you must find the value of r
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Example (cont)
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Example
Given that ? is an acute angle and
, find the exact value of the five remaining
trig functions of ?.

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Example
Find the value of tan ? given csc ? 1.02,
where ? is an acute angle. Give answer to three
significant digits.
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Special Right Triangles
The 45º- 45º- 90º Triangle

Ratio of the sides
Find the exact values of the six trig functions
for 45? sin 45? csc 45? cos 45? sec
45? tan 45? cot 45?
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Special Right Triangles
The 30º- 60º- 90º Triangle

Ratio of the sides
Find the exact values of the six trig functions
for 30? sin 30? csc 30? cos 30? sec
30? tan 30? cot 30?
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Special Right Triangles
The 30º- 60º- 90º Triangle

Ratio of the sides
Find the exact values of the six trig functions
for 60? sin 60? csc 60? cos 60? sec
60? tan 60? cot 60?
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Using the calculator to evaluate trig functions
Make sure the MODE is set to the correct unit of
angle measure (i.e. Degree vs.
Radian) Example Find to
three significant digits.

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Using the calculator to evaluate trig functions
For reciprocal functions, you may use the
reciprocal button ? , but DO NOT USE THE INVERSE
FUNCTIONS (e.g. ?)! Example 1. Find
2. Find (to 3
significant dig) (to 4
significant dig)

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Angles and Accuracy of Trigonometric Functions
Measurement of Angle to Nearest Accuracy of Trig Function
1 2 significant digits
0. 1 or 10 3 significant digits
0. 01 or 1 4 significant digits
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The inverse trig functions give the measure of
the angle if we know the value of the function.
NotationThe inverse sine function is denoted
as sin-1x or arcsinx. It means the angle whose
sine is x. The inverse cosine function is
denoted as cos-1x or arccosx. It means the angle
whose cosine is x. The inverse tangent function
is denoted as tan-1x or arctanx. It means the
angle whose tangent is x.
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Example
is the angle whose sine is
Think of the related statement
? must be 30,
therefore
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Examples Given that 0 ? 90, use an inverse
trig functions to find the value of ? in degrees.
To nearest 0.1?
To 2 sig. dig.
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Examples Given that 0 ? 90, use an inverse
trig functions to find the value of ? in degrees.
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Example Solve the right triangle with the
indicated measures.
Solution
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Example
Solution
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Angle of Elevation and Angle of Depression
The angle of elevation for a point above a
horizontal line is the angle formed by the
horizontal line and the line of sight of the
observer at that point. The angle of depression
for a point below a horizontal line is the angle
formed by the horizontal line and the line of
sight of the observer at that point.
Horizontal line
Angle of depression
Angle of elevation
Horizontal line