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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

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

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.

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.

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.

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

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

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

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.

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

Applications of Trigonometric Functions

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