Transcript ch. 2

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Chapter 2
Acute Angles and
Right Triangles
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2.1
Trigonometric Functions of
Acute Angles
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Right-triangle Based Definitions of
Trigonometric Functions

For any acute angle A in standard position.
y side opposite
sin A  
r
hypotenuse
x side adjacent
cos A  
r
hypotenuse
y side opposite
tan A  
x side adjacent
x side adjacent
cot A  
.
y side opposite
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r
hypotenuse
csc A  
y side opposite
r
hypotenuse
sec A  
x side adjacent
Slide 2-4
Example: Finding Trig Functions of
Acute Angles

Find the values of sin A, cos A, and tan A in the
right triangle shown.
48
A
C
20
52
B
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Slide 2-5
Cofunction Identities

For any acute angle A,

sin A = cos(90  A)
csc A = sec(90  A)

tan A = cot(90  A)
cos A = sin(90  A)

sec A = csc(90  A)
cot A = tan(90  A)
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Slide 2-6
Example: Write Functions in Terms of
Cofunctions

Write each function in
terms of its cofunction.

a) cos 38
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
b) sec 78
Slide 2-7
Example: Solving Equations

Find one solution for the equation
cot(4  8 )  tan(2  4 ) .
o
o
Assume all angles are acute angles.
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Slide 2-8
Example: Comparing Function Values

Tell whether the statement is true or false.
sin 31 > sin 29

In the interval from 0 to 90, as the angle
increases, so does the sine of the angle, which
makes sin 31 > sin 29 a true statement.
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Slide 2-9
Special Triangles

30-60-90 Triangle
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
45-45-90 Triangle
Slide 2-10
14, 40, 54, 56, 62
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Slide 2-11
Function Values of Special Angles

sin 
30
1
2
3
2
3
3
3
2 3
3
2
45
2
2
2
2
1
1
2
2
60
3
2
1
2
3
3
3
2
2 3
3
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cos 
tan 
cot 
sec 
csc 
Slide 2-12
2.2
Trigonometric Functions of
Non-Acute Angles
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Reference Angles

A reference angle for an angle  is the positive
acute angle made by the terminal side of angle 
and the x-axis.
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Slide 2-14
Example: Find the reference angle for
each angle.


a) 218
Positive acute angle
made by the terminal side
of the angle and the xaxis is 218  180 = 38.
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
1387
Slide 2-15
Example: Finding Trigonometric Function
Values of a Quadrant Angle

Find the values of the
trigonometric functions for
210.
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Slide 2-16
Finding Trigonometric Function Values
for Any Nonquadrantal Angle 

Step 1


Step 2
Step 3

Step 4
If  > 360, or if  < 0, then find a
coterminal angle by adding or subtracting
360 as many times as needed to get an
angle greater than 0 but less than 360.
Find the reference angle '.
Find the trigonometric function values for
reference angle '.
Determine the correct signs for the values
found in Step 3. (Use the table of signs in
section 5.2, if necessary.) This gives the
values of the trigonometric functions for
angle .
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Slide 2-17
Example: Finding Trig Function Values
Using Reference Angles


Find the exact value of
each expression.
cos (240)
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Slide 2-18
Example: Evaluating an Expression
with Function Values of Special Angles

Evaluate cos 120 + 2 sin2 60  tan2 30.
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Slide 2-19
Example: Using Coterminal Angles


Evaluate each function by first expressing the
function in terms of an angle between 0 and
360.
cos 780
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Slide 2-20
45, 48, 49, 8, 10, 12, 14, 36, 38
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Slide 2-21
2.3
Finding Trigonometric Function
Values Using a Calculator
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Function Values Using a Calculator




Calculators are capable of finding trigonometric
function values.
When evaluating trigonometric functions of
angles given in degrees, remember that the
calculator must be set in degree mode.
To check if your calculator is in degree mode
enter sin 90. The answer should be 1.
Remember that most calculator values of
trigonometric functions are approximations.
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Slide 2-23
Example: Finding Function Values with
a Calculator

a) sin 38 24
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
b) cot 68.4832 
Slide 2-24
Angle Measures Using a Calculator


Graphing calculators have three inverse
functions.
If x is an appropriate number, then sin 1 x,cos1 x,
or tan -1 x gives the measure of an angle whose
sine, cosine, or tangent is x.
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Slide 2-25
Example: Using Inverse Trigonometric
Functions to Find Angles

Use a calculator to find an angle  in the
interval [0 ,90 ] that satisfies each condition.
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Slide 2-26
Example: Using Inverse Trigonometric
Functions to Find Angles continued

sec  2.486879
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Slide 2-27
50, 58
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Slide 2-28
2.4
Solving Right Triangles
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Significant Digits for Angles


A significant digit is a digit obtained by actual measurement.
Your answer is no more accurate then the least accurate
number in your calculation.
Number of
Significant Digits
Angle Measure to Nearest:
2
Degree
3
Ten minutes, or nearest tenth of a degree
4
Minute, or nearest hundredth of a degree
5
Tenth of a minute, or nearest thousandth of
a degree
Example
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2
Slide 2-30
Example: Solving a Right Triangle,
Given an Angle and a Side

Solve right triangle ABC, if A = 42 30' and c = 18.4.
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Slide 2-31
Example: Solving a Right Triangle
Given Two Sides

Solve right triangle ABC if a = 11.47 cm and c = 27.82 cm.
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Slide 2-32
Definitions

Angle of Elevation: from
point X to point Y (above
X) is the acute angle
formed by ray XY and a
horizontal ray with
endpoint X.
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
Angle of Depression:
from point X to point Y
(below) is the acute angle
formed by ray XY and a
horizontal ray with
endpoint X.
Slide 2-33
Solving an Applied Trigonometry Problem

Step 1

Step 2

Step 3
Draw a sketch, and label it with the
given information. Label the quantity to
be found with a variable.
Use the sketch to write an equation
relating the given quantities to the
variable.
Solve the equation, and check that
your answer makes sense.
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Slide 2-34
46
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Slide 2-35
Example: Application


The length of the shadow of a tree 22.02 m tall is
28.34 m. Find the angle of elevation of the sun.
Draw a sketch.
22.02 m
B
28.34 m
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Slide 2-36
2.5
Further Applications of
Right Triangles
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Bearing

Other applications of right triangles involve
bearing, an important idea in navigation.
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Slide 2-38
Example

An airplane leave the airport flying at a bearing of N 32E
for 200 miles and lands. How far east of its starting point
is the plane?
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Slide 2-39
16, 22
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Slide 2-40
Example: Solving a Problem Involving
Angles of Elevation

Sean wants to know the height of a Ferris wheel.
From a given point on the ground, he finds the
angle of elevation to the top of the Ferris wheel
is 42.3 . He then moves back 75 ft. From the
second point, the angle of elevation to the top of
the Ferris wheel is 25.4 . Find the height of the
Ferris wheel.
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Slide 2-41
33
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Slide 2-42