7.1
Right Triangle Trigonometry
A triangle in which one angle is a right angle is
called a right triangle. The side opposite the
right angle is called the hypotenuse, and the
remaining two sides are called the legs of the
triangle.
c
b
90
a
Find the value of each of the six trigonometric
functions of the angle
c = Hypotenuse = 13
12 13
b = Opposite = 12
Adjacent
a Adjacent = 5
b Opposite = 12
c Hypotenuse = 13
c
a
b
To solve a right triangle means to
find the missing lengths of its sides
and the measurements of its angles.
c
4
35
b
h 25
h = 23.49
70
7.2
The Law of Sines
If none of the angles of a triangle is a right
angle, the triangle is called oblique.
All angles are acute
Two acute angles, one obtuse angle
To solve an oblique triangle means to
find the lengths of its sides and the
measurements of its angles.
A
S
A
ASA
S
A A
SAA
CASE 1: ASA or SAA
S
A
S
CASE 2: SSA
S
A
S
CASE 3: SAS
S S
S
CASE 4: SSS
The Law of Sines is used to solve
triangles in which Case 1 or 2
holds. That is, the Law of Sines
is used to solve SAA, ASA or SSA
triangles.
Theorem Law of Sines
7.3
Law of Cosines
We use the Law of Sines to solve CASE 1
(SAA or ASA) and CASE 2 (SSA) of an
oblique triangle. The Law of Cosines is
used to solve CASES 3 and 4.
CASE 3: Two sides and the included
angle are known (SAS).
CASE 4: Three sides are known (SSS).
Theorem Law of Cosines
Remember to give alternate form of law of cosines!
7.4
Area of a Triangle
Theorem
The area A of a triangle is
where b is the base and h is the altitude
drawn to that base.
Theorem
The area A of a triangle equals one-half the
product of two of its sides times the sine of
its included angle.
Theorem Heron’s Formula
The area A of a triangle with sides a,
b, and c is
Find the area of a triangle whose sides are
5, 8, and 11.
Additional Examples
Page 561: 25, 27, and 29
7.5
Simple Harmonic Motion;
Damped Motion;
Combining Waves
Simple harmonic motion is a
special kind of vibrational motion
in which the acceleration a of the
object is directly proportional to
the negative of its displacement d
from its rest position. That is,
a = -kd, k > 0.
Theorem Simple Harmonic Motion
An object that moves on a coordinate axis
so that its distance d from the origin at
time t is given by either
The frequency f of an object in simple
harmonic motion is the number of
oscillations per unit of time. Thus,
Suppose an object is attached to a
pendulum and is pulled a distance 7 meters
from its rest position and then released. If
the time for one oscillation is 4 seconds,
write an equation that relates the distance d
of the object from its rest position after
time t (in seconds). Assume no friction.
Suppose that the distance d (in centimeters)
an object travels in time t (in seconds)
satisfies the equation
(a) Describe the motion of the object.
Simple harmonic
(b) What is the maximum displacement
from its resting position?
A = |-15| = 15 centimeters.
Suppose that the distance d (in centimeters)
an object travels in time t (in seconds)
satisfies the equation
d 15sin 4t
(c) What is the time required for one
oscillation?
Period :
(d) What is the frequency?
frequency oscillations per second.
Theorem Damped Motion
The displacement d of an oscillating object
from its at rest position at time t is given by
where b is a damping factor (damping
coefficient) and m is the mass of the
oscillating object.
Suppose a simple pendulum with a bob of
mass 8 grams and a damping factor of 0.7
grams/second is pulled 15 centimeters to
the right of its rest position and released.
The period of the pendulum without the
damping effect is 4 seconds.
(a) Find an equation that describes the
position of the pendulum bob.
(b) Using a graphing utility, graph the
function.
(c) Determine the maximum displacement
of the bob after the first oscillation.
Assignment
Page 561: 10, 18, 26, 30, and 40
Page 571: 16, 18, and 28