1 right triangle trigonometry. 2 angles trigonometry: measurement of triangles angle measure
TRANSCRIPT
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Right Triangle Trigonometry
2
Angles
Trigonometry: measurement of triangles
Angle Measure
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In this section, we will be studying special ratios of the sides of a right triangle, with respect to angle, .
These ratios are better known as our six basic trig functions:
Sine of
Cosine of
Tangent of
Cosecant of
Secant of Cotangent of
Trigonometric Functions
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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 .
The sides of a right triangle
Side adjacent to
S
ide
op
po
site
Hypotenuse
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Definitions of the Six Trigonometric Functions
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To remember the definitions of Sine, Cosine and Tangent, we use the acronym :
“SOH CAH TOA”
Definitions of the Six Trigonometric Functions
O A O
H HS C
AT
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Find the exact value of the six trig functions of :
Example
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9 First find the length of the hypotenuse using the Pythagorean Theorem.
2
2
2
hyp
hyp
hyp
hyp
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Example (cont)
5
9
106
So the six trig functions are:
sin
cos
tan
opp
hyp
adj
hyp
opp
adj
csc
sec
cot
hyp
opp
hyp
adj
adj
opp
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Given that is an acute angle and , find the exact value of the six trig functions of .
Example
12cos
13
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Find the value of sin given cot = 0.387, where is
an acute angle. Give answer to three significant digits.
Example
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The 45º- 45º- 90º Triangle
Special Right Triangles
1
12
45º
45º
Find the exact values & decimal approximations (to 3 sig digits) of the six trig functions for 45
sin 45 = csc 45 =
cos 45 = sec 45 =
tan 45 = cot 45 =
Ratio of the sides:
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The 30º- 60º- 90º Triangle
Special Right Triangles
Find the exact values & decimal approximations (to 3 sig digits) of the six trig functions for 30
sin 30 = csc 30 =
cos 30 = sec 30 =
tan 30 = cot 30 =
1
3
60º
30º
2
Ratio of the sides:
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The 30º- 60º- 90º Triangle
Special Right Triangles
Find the exact values & decimal approximations (to 3 sig digits) of the six trig functions for 60
sin 60 = csc 60 =
cos 60 = sec 60 =
tan 60 = cot 60 =
1
3
60º
30º
2
Ratio of the sides:
14
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MAKE SURE THE MODE IS SET TO THE CORRECT UNIT OF ANGLE MEASURE (i.e. Degree vs. Radian)
Example:
Find to three significant digits.
Using the calculator to evaluate trig functions
tan 46.2
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For reciprocal functions, you may use the button, but DO NOT USE THE INVERSE FUNCTIONS (e.g. )!
Example:
1. Find 2. Find
(to 3 significant dig) (to 4 significant dig)
Using the calculator to evaluate trig functions
csc73.2 cot 11.56
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THE INVERSE TRIG FUNCTIONS GIVE THE MEASURE OF THE ANGLE IF WE KNOW THE VALUE OF THE FUNCTION.
Notation:The inverse sine function is denoted as sin-1x.
It means “the angle whose sine is x”.
The inverse cosine function is denoted as cos-1x. It means “the angle whose cosine is x”.
The inverse tangent function is denoted as tan-1x. It means “the angle whose tangent is x”.
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Examples
Evaluate the following inverse trig functions using the
calculator. Give answer in degrees.
1 11. tan 1.372 2. sin 0.64
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Examples
Evaluate the following inverse trig functions using the
calculator. Give answer in degrees.
1 13. tan 1 4. cos 0.541
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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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Example
Solve for y:
Solution:
y
52º
9.6
Since you are looking for the side adjacent to 52º and are given the hypotenuse, you should use the _____________ function.
WARNING: Make sure your MODE is set to “Degree”
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Example
Solve the right triangle with the indicated measures.
1. 40.7 8.20A a in
Solution
A= 40.7°
C B
b c
a=8.2”
Answers:
B
b
c
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Example
2. 25 35a c A
C B
b c=35
a=25
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Example
3. Find the altitude of the isosceles triangle below.
36°
8.6 m
36°
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Example
4. Solve the right triangle with 8.60 11.25a cm b cm
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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
Horizontal line
Angle of elevation
Angle of depression
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Example
A guy wire of length 108 meters runs from the top of an
antenna to the ground. If the angle of elevation of the top
of the antenna, sighting along the guy wire, is 42.3° then
what is the height of the antenna? Give answer to three
significant digits.
Solution
108 m
42.3°
y
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