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,cos1 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 32E
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