Download Day 5: Blank Notes + HW Assignment

Section 2.4: Trigonometric Identities
It will be important to know these now as we will work on them
down the road. First, some reminders:
What we will add:
Pythagorean Identities
Some definitions first.
In class derivation:
We can “divide out” sine or cosine to get two more identities.
In Class Example (see example 3 on pg 114):
In Class Example (see example 5 on pg 115):
Section 3.1: Trigonometric Identities
Instead of using degrees, we can also look at angles using a
different system. Let’s define some stuff.
A central angle is an angle that has its vertex at the center of a
circle.
So the technical definition of a radian is the length of the arc on the
circle divided by the length of the radius. Here are some key angles
that lead to some key radians.
When the length of the arc on the circle,
(aka “s”) equals the length of the radius,
(aka “r”), s/r becomes 1.
A radian is about 57°.
The length of the entire circle (technically the circumference)
depends on the radius. A circle with radius r will have a
circumference of 2πr. So the radian measure of all 360° of a circle
would be:
360° in radians =
௦
௥
=
ଶగ௥
௥
= 2ߨ
In Class Example (see examples 1 and 2 on pg 131):
What is the measure (in radians) of a central angle ϴ that intercepts an arc
of length 12 millimeters on a circle with radius 4 centimeters?
A List of the Metric Prefixes
Prefix
Symbol Numerical
kilo
k
1,000
hecto
h
100
deca
da
10
no prefix: 1
100
deci
d
0.1
centi
c
0.01
milli
m
0.001
Because radians are unitless, the word radians (or rad) is often omitted. If
an angle measure is given simply as a real number, then radians are
implied.
WORDS
The measure of ϴ is 4 degrees. → ϴ = 4°
The measure of ϴ is 4 radians. → ϴ = 4
Converting Between Degrees and Radians
Recall: 360° in radians
௦
௥
=
ଶగ௥
௥
= 2ߨ
If 360°=2 π, then it stands to reason that 180°= π and 90°= π/2 and
45°= π/4
Notice: 180°= π gives us a way to convert between radians and
degrees.
In Class Example (see examples 3-6 on pgs 133-134):
a) Convert 460° to radians.
b) Convert 7 radians to degrees. Use the π key!
Reference Page
In Class Example (see examples 7-9 on pgs 135-137):
Find the reference angle for
ହగ
ଷ
and the exact value of cosine.
Show all work. Rationalize denominators where needed.
Section 2.4: 8, 16, 17, 21, 22, 27, 32, 35, 39, 42, 57
Problem hints:
2.4.32: find cosine first to calculate tangent.
2.4.39 and 2.4.42: set up triangle in the right quadrant and calculate
hypotenuse
Section 3.1: 7, 8, 20, 21, 34, 35, 42, 43, 48, 49, 54, 55, 68*, 74*,
78*, 80*, 92♦
*For these problems, state the reference angle and quadrant as well.
♦
For this problem, they’re basically saying that 45 minutes is equal to 2π. So what
fraction of 2π would 25 minutes give you? Show work and fractions.