Solving Right Triangles

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Solving Right Triangles
DateTopicApplications of Trigonometric
Functions(4.8)
Solve the right triangle shown.
Find the measure of angle B. B 90º A B
90º 34.5º B 55.5º We have a known angle, an
unknown opposite side, and a known adjacent side,
use ? the tangent function. tan 34.5º
a/10.5 10.5 tan 34.5 º a 7.22 a
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Text Example cont.
We need to find c.
We have a known angle, a known adjacent side, and
an unknown hypotenuse, we use ? the cosine
function. cos 34.5 º 10.5/c
c 10.5/cos 34.5 º c
12.74 So, B 55.5º, a 7.22, and c 12.74
3
Text Example
N
Use the figure to find a. the bearing from O to
B. b. the bearing from O to A.
40º
B
A
20º
a.
W
E
O
We need the angle between the ray OB and the
north-south line through O. The measurement of
this angle is given to be 40º. This angle is
west of that line. Thus, the bearing from O to B
is N 40º W
75º
D
C
25º
S
We need the angle between the ray OA and the
north-south line through O. This angle measures
90º 20º, or 70º. This angle is east of the
north-south line. Thus the bearing from O to A
is N 70º E
b.
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Text Example
N
Use the figure to find c. the bearing from O to
C. d. the bearing from O to D.
40º
B
A
20º
c.
W
E
O
We need the angle between the ray OC and the
north-south line through O. This angle measures
90º 75º, or 15º and is west of the north-south
line. Thus, the bearing from O to C is
S 15º W
75º
D
C
25º
S
We need the angle between the ray OD and the
north-south line through O. This angle measures
25º. This angle is east of the north-south line.
Thus the bearing from O to A is S 25º E
d.
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Complete Student Checkpoint
You leave the entrance to a system of hiking
trails and hike 2.3 miles on a bearing of S 31º
W. Then the trails turns 90º clockwise and you
hike 3.5 miles on a bearing of N 59º W. At that
time a. How far are you from the entrance to
the trail system?
Use the pythagorean theorem to find the distance.
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Complete Student Checkpoint
b. What is your bearing from the entrance to the
trail system?
Your bearing from the entrance to the trail
system is
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Simple Harmonic Motion

• An object that moves on a coordinate axis is in
  simple harmonic motion if its distance from the
  origin, d, at time t is given by either
• d a cos ? t or d a sin ? t
• The motion has amplitude a, the maximum
  displacement of the object from its rest
  position. The period of the motion is 2?/? ,
  where ? gt 0. The period gives the time it takes
  for the motion to go through one complete cycle.
  It has frequency f given by f ?/2?, ?gt0.
  Equivalently, f 1/period

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Example

• An object in simple harmonic motion has a
  frequency of 1/4 oscillation per minute and an
  amplitude of 8 ft. Write an equation in the form
  for the objects simple
  harmonic motion.

a 8 and the period is 4 minutes since it
travels 1/4 oscillation per minute
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Example

• A mass moves in simple harmonic motion described
  by the following equation, with t measured in
  seconds and d in centimeters. Find the maximum
  displacement, the frequency, and the time
  required for one cycle.

The frequency is
1/6 cm per second.
The maximum displacement is 8 cm, since a 8
The time required for one cycle is
6 seconds.
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Complete Student Checkpoint
A ball on a spring is pulled six inches below
its rest position and then released. The period
for the motion is 4 seconds. Write the equation
for the balls simple harmonic motion.
a -6
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Applications of Trigonometric Functions