Lesson 14:
Sine, Cosine, and Tangent, Angles of Elevation and Depression, Rectangular and Polar Coordinates, Coordinate Conversion
Warm Up:
Identify the Parts of the Right Triangle in relation to Angle A:
A
Vocabulary:
Opposite (side): The side directly across from selected angle
Adjacent (side): the side next to the selected angle that is not the hypotenuse
Hypotenuse: The side of a right triangle directly across from the right angle
Sine:o/h
Cosine:a/h
Tangent:o/a
Unit Vector: A vector with a length of one.
Sine, Cosine, Tangent:
Sin A: Opposite/Hypotenuse
Cos A: Adjacent/Hypotenuse
Tan A: Opposite/Adjacent
SohCahToa
A
Opposite
Adjacent
Hypotenuse
Sine, Cosine, Tangent(2):
Uses of Sine, Cosine, and Tangent:
Use Sin and Tan to find x and y:
8/y=sin 15’ 8/x=tan 15’
8=ysin 15’ 8=xtan 15’
8/sin 15’=y 8/tan 15’=x
y~~30.9’ x~~29.9’
Use cos to find j, and the pythagorean theorem to find k:
cos j=8/13 13^2-8^2=k^2
cos j~~0.61538461538….. 169-64=k^2
Inverse cos= cos^-1 0.61538461538~~52.0 105=k^2
j~~52’ 105^½=k
k~~10.2
15 ‘
8
x
y
j
8
13
k
Note: Be SURE your calculator is set to degrees
~~: Approximately equal to
Angles of Depression/Elevation:
Measured upward from horizontal
Measured downward from horizontal
Horizontal
Line of sight
Angle of Elevation
Horizontal
Line of sight
Angle of Depression
Angle of Elevation
Angle of Depression
Angles of Depression/Elevation(2):
A 5th tall man is 20 ft away from a tree, and measures the angle of elevation to the top of the tree is 30’. How tall is the tree?
Solve for y using tan:
y/20=tan 30’
y=20tan 30’
y~~11.55
Now add the man’s height as well!
5+11.55=16.55
Height of the tree:16.55ft
5 ft
20 ft
30’
y
Angles of Depression/Elevation(3):
An unmanned drone is flying at an altitude of 2000ft. The village that is about to be blown up is at an angle of depression of 30’. What is the slant length from the drone to the village?
2000ft
30’
60’
Angle of Depression: 30’
Angle A: 60’
Find s using cos:
cos 60’ = 2000/s
s=2000/cos 60’
s~~4000
s
=4000
Regular/Polar Coordinates:
Regular Coordinates: (-8, 6) = 8 to the left of the origin, 6 up from the origin
Written without the parentheses: -8i^ + 6j^
i^=Used to indicate positive x direction. Unit Vector.
j^=Used to indicate positive y direction. Unit Vector.
Ordered Pair: (-8,6)
Vector Form: -8i^+6j^
8
-8
-6
6
(-8,6) or -8i^+6j^
Regular/Polar Coordinates(2):
Polar Coordinates: Uses a distance and an angle
Positive Angles: Counterclockwise from positive x axis
Negative Angles: Clockwise from positive x axis
-200’
7
Can be written in two ways:
(7,-220’) or 7 -200’
Note: (-7, 290’) ,(7, 160’) and (-7,-20) get you to the same place!
160’
Coordinate Conversation:
To convert from polar to regular coordinates:
1. Draw a diagram
2. Determine the length of the hypotenuse
3. Measure the angle
4. Measure the polar angle from the positive x axis
Coordinate Conversation(2):
Convert (2,3)/2i^ + 3j^ to polar coordinates.
(3,2)
(3,0)
(0,0)
r
2
r^2=3^2 + 2^2
r^2=13
r=13^1/2
tan x = ⅔=0.6667
tan^-1=33.69’
33.69’
(13^½,-326.31’)
Other ways to write it:
(13^½, 33.69’)
(-13^½, -146.31’)
(-13^½, -213.69)
=13^1/2
-326.31’
3
Coordinate Conversation (3):
Convert (-13, -253’) to Rectangular Form:
Step 1: Draw the angle first
-253’
73’
Step 2: Go down 13 units from the origin in the direction c is pointing
c
13
a
b
a=13cos 73’
a~~3.80
b=13sin 73’
b~~12.43
(3.80, -12.43) or 3.80i^ - 12.43j^
73’
Practice 1:
Find x and y:
8
9
x’
y
Practice 2:
A 6ft tall man is 50 feet from a streetlight, and he measures the angle of elevation at 20’. How tall is the streetlight?
6ft
20’
50 ft
Practice 3:
Convert (2,2) or 2i^ + 2j^ to Polar Coordinates.
(2,2)
Summary:
Sine: o/h
Cosine: a/h
Tan: o/a
Angle of Elevation: Measured upward from the horizontal
Angle of Depression: Measured downward from the horizontal
Regular Coordinates: Uses two numbers, one indicating the distance of the point to the right or left of the origin, the other the distance above or below the axis
Polar Coordinates: Uses a distance and an angle
Homework:
Pg 113
4-17 all
Sources Cited:
Saxon, John H., Jr. "Lesson 14." Advanced Mathematics: An Incremental Development. 2nd ed. Norman, OK: Saxon, 1997. 108-112. Print.
"Unit Vector." Wolfram MathWorld. N.p., n.d. Web. 28 Sept. 2013.