Right Triangle Trigonometry
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Transcript Right Triangle Trigonometry
Warm-up 10/28
Evaluate the following. Give exact values when possible:
1
arcsin
2
1
arccos
2
1
arctan
2
4.8 Trigonometric Applications
and Models 2014
Objectives:
•Use right triangles to solve real-life problems.
•Use directional bearings to solve real-life problems.
•Use harmonic motion to solve real-life problems.
Terminology
• Angle of elevation –
angle from the
horizontal upward to
an object.
• Angle of depression –
angle from the
horizontal downward
to an object.
Object
Horizontal
Angle of
elevation
Observer
Angle of
depression
Observer
Horizontal
Object
Precalculus
4.8 Applications and Models
3
Example
• Solve the right triangle for
all missing sides and angles.
Example – Solving Rt. Triangles
At a point 200 feet from
the base of a building,
the angle of elevation to
the bottom of a smokestack
is 35°, and the angle of
elevation to the top of the
smokestack is 53°. Find the
height of the smokestack.
Precalculus
4.8 Applications and Models
6
You try:
• A swimming pool is 20 meters long and 12 meters
wide. The bottom of the pool is slanted so that the
water depth is 1.3 meters at the shallow end and 4
meters at the deep end, as shown. Find the angle of
depression of the bottom of the pool.
7.69°
Trigonometry and Bearings
• In surveying and navigation, directions are
generally given in terms of bearings. A bearing
measures the acute angle that a path or line of
sight makes with a fixed north-south line.
N 35E
N
N
S 70W
35°
W
E
W
E
70°
S
S
Trig and Bearings
• You try. Draw a bearing of:
N800W
S300E
N
N
W
E
S
W
E
S
Trig and Bearings
• You try. Draw a bearing of:
N800W
S300E
800
300
Example – Finding Directions Using Bearings
• A hiker travels at 4 miles per hour at a
heading of S 35° E from a ranger station.
After 3 hours how far south and how far east
is the hiker from the station?
Precalculus
4.8 Applications and Models
11
20sin(36o)
Example – Finding Directions Using Bearings
A ship leaves port at noon and heads due west at 20 knots, or 20
nautical miles (nm) per hour. At 2 P.M. the ship changes course to N
54o W. Find the ship’s bearing and distance from the port of
departure at 3 P.M.
d
a
θ
b
o)
20cos(36
a
20
o
20sin(36 ) a
b
cos(36o )
20
o
20cos(36 ) b
sin(36o )
40 nmfor 2
20 nmph
hrso ) 40
20cos(36
20sin(36o )
tan( )
20cos(36o ) 40
o
20sin(36
)
1
tan
o
20cos(36
)
40
11.819o
Bearing: N
78.181o W
20sin(36o)
A ship leaves port at noon and heads due west at 20
knots, or 20 nautical miles (nm) per hour. At 2 P.M. the
ship changes course to N 54o W. Find the ship’s bearing
and distance from the port of departure at 3 P.M.
d
a
θ
b
o)
20cos(36
a
20
o
20sin(36 ) a
b
cos(36o )
20
o
20cos(36 ) b
sin(36o )
40 nmfor 2
20 nmph
hrso ) 40 Bearing: N 78.181o W
20cos(36
20sin(36 )
20sin(36 )
o
o
20 cos(36 ) 40
2
20 cos(36 ) 40 d 2
2
o
o
2
2
d
57.397 nm d
Two lookout towers are 50 kilometers apart. Tower A is due west of
tower B. A roadway connects the two towers. A dinosaur is spotted
from each of the towers. The bearing of the dinosaur from A is N 43o
E. The bearing of the dinosaur from tower B is N 58o W. Find the
distance of the dinosaur to the roadway that connects the two towers.
h
43o
58o
47o
A
h
tan(47 )
x
x tan(47o ) h
o
32o
x
50– x
h
tan(32 )
50 x
50 x tan(32o ) h
o
B
h
47o
A
32o
x
x tan(47o ) h
50 tan(32o )
o
tan(47
)h
o
o
tan(47 ) tan(32 )
19.741 h
19.741
50– x
B
50 x tan(32o ) h
x tan(47o ) 50 x tan(32o )
x tan(47o ) 50 tan(32o ) x tan(32o )
x tan(47o ) x tan(32o ) 50 tan(32o )
x tan(47o ) tan(32o ) 50 tan(32o )
50 tan(32o )
x
km
tan(47o ) tan(32o )
Homework
4.8 p 326
1, 5, 9, 17-37 Odd
Quiz tomorrow on sections 4.5,4.6, and 4.7
Precalculus
4.8 Applications and Models
19
4.8 Trigonometric Applications
and Models Day 2
Objectives:
•Use harmonic motion to solve real-life problems.
Terminology
• Harmonic Motion – Simple vibration,
oscillation, rotation, or wave motion. It can
be described using the sine and cosine
functions.
• Displacement – Distance from equilibrium.
Precalculus
4.8 Applications and Models
22
Simple Harmonic Motion
• A point that moves on a coordinate line is in
simple harmonic motion if its distance d from the
origin at time t is given by
d a sin t
a amplitude
or
d a cos t
2
period
frequency
2
where a and ω are real numbers (ω>0)
and frequency is number of cycles per unit of time.
Precalculus
4.8 Applications and Models
23
Simple Harmonic Motion
10 cm
10 cm
0 cm
0 cm
10 cm
10 cm
Precalculus
4.8 Applications and Models
24
Example – Simple Harmonic Motion
Given this equation for simple harmonic motion
3
d 6 cos
t
4
Find:
a) Maximum displacement 6
b) Frequency 83 cycle per unit of time
c) Value of d at t=4 6
d) The least positive value of t when d=0
Precalculus
4.8 Applications and Models
2
t
3
25
You Try – Simple Harmonic Motion
A mass attached to a spring vibrates up and down in
simple harmonic motion according to the equation
d 4sin
2
t
Find:
a) Maximum displacement 4
1
b) Frequency 4 cycle per unit of time
c) Value of d at t 2 0
d) 2 values of t for which d=0 0 and 2
Example – Simple Harmonic Motion
A weight attached to the end of a spring is pulled down
5 cm below its equilibrium point and released. It takes
4 seconds to complete one cycle of moving from 5 cm
below the equilibrium point to 5 cm above the
equilibrium point and then returning to its low point.
• Find the sinusoidal function that best represents
the motion of the moving weight. f t 5cos t
2
• Find the position of the weight 9 seconds after it is
released. 0
You Try – Simple Harmonic Motion
A buoy oscillates in simple harmonic motion as waves
go past. At a given time it is noted that the buoy
moves a total of 6 feet from its high point to its low
point, returning to its high point every 15 seconds.
• Write a sinusoidal function that describes the
motion of the buoy if it is at the high point at t=0.
2
f t 3cos
t
15
• Find the position of the buoy 10 seconds after it is
released. 3
2
Homework:
4.8 Applications and Models Worksheet
(Bearings and Harmonic Motion)
Test next Tuesday.