Download 3.1 Trig Functions of Non-Acute Angles

Math 170
Trigonometry Lecture Notes
Chapter 3
3.1 Trig Functions of Non-Acute Angles
Video: Trig Functions of Non-Acute Angles (7:39)
Reference Angles:
∙Acute angle made with the ________ -axis
∙ symbol for reference angle: _____
To find the Reference Angle:
a) Find a coterminal angle between __________ and __________ by adding or subtracting 360.
b) Draw the coterminal angle and reference angle.
Find the reference angles for:
1) 1216⁰
2) - 986⁰
3) 575⁰
Theorem: A trigonometric function of an angle and its
reference angle are the same, except, perhaps, for a
difference in sign.
sin 150o = ____ = sin ____
sin 210o = _____ = sin ____
(See HW video for more practice on Reference angles)
Definition: The trigonometric functions of an angle and any angle coterminal to it are always equal. (pg 118)
for any Integer k:
sin (ϴ + 360ok) = sin ___
cos(ϴ + 360ok) = cos ____
Example:
1. cos 495o = cos _____ ?
2. Find ϴ to nearest degree if sin ϴ = -0.5592 and ϴ terminates in QIII
with 0o < ϴ < 360o
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Math 170
Trigonometry Lecture Notes
Chapter 3
“Special Angles”: Any angle whose reference angle is 30, 45, or 60
* We can find “Exact Values” for all the Trig functions of these angles
* “Exact Values” will have radicals, they are not decimal approximations from the calculator.
Find exact value of each of the following:
1) tan 315⁰
2) cos (-1860⁰)
3) sin (750⁰)
Use your knowledge of Trig, not a calculator, to find if the following are True or False:
1) sin 30˚ + sin 60˚ = sin(30˚ + 60˚)
2) sin 120˚ = sin 150˚ – sin 30˚
CA 3.1 #5, 15, 27, 57, 61, 75
3.2 Radian Measure
Definition:
ϴ (in radians) =
s
,
r
for circle with radius r and
s = arc length of the arc cut off by ϴ
Definition: A central angle that cuts off of an arc equal in length
to the radius of the circle has a measure of ___ radian
What is the formula for the Circumference of a Circle? C = _____
What happens if the radius = 1?
•
In Radians, an angle measurement, there are ___________ radians in a circle.
•
Converting between Degrees and Radians: __π radians__ = ______________
•
1 radian ≈ 57.3o
Radian measurements will not have any unit symbol. So if we say: “Find sin 2,” we mean 2 radians.
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Math 170
Trigonometry Lecture Notes
Chapter 3
Draw the following in standard position:
Convert from Degrees to Radians and find the reference angle in both degrees and radians:
1) 45⁰
2) 60⁰
3) 90⁰
Convert from Radians to Degrees and find the reference angle in both radians and degrees.
1)
2π
3
2)
3π
4
3)
5π
6
CA: 3.2 # 19, 11, 68, 73, 77, 83
3.3 Approximating trig functions of values in radians
** Be sure to change to Radian Mode **
a) cot 3.79 ← note no o sign so this
is in radians
b) sec 0.12
c) csc θ = 8.5
3.3 The Unit Circle & Circular Functions
On the Unit Circle we have r = 1
→ Note: cos ϴ = x/r = x/1 = x
→ Given arc of length t, such that ϴ = t/r = t/1 = t
We can actually calculate: cos t , since t = ϴ
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Math 170
Trigonometry Lecture Notes
Chapter 3
Find the patterns:
Function
P(x,y)
radius, r
Unit Circle,
r=1
Function
ϴ=t
sin θ
csc θ
cos θ
sec θ
tan θ
cot θ
P(x,y)
radius, r
Unit Circle,
r=1
ϴ=t
Use “Unit Circle Handout” from HW video: How to remember the unit circle cc (12:02)
(pg 141)
Domains of Circular Functions
Ranges of Circular functions
sin t, cos t
tan t, sec t
cot t, csc t
CA: 3.3 # 26, 29, 78, 72, 32
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Math 170
Trigonometry Lecture Notes
Chapter 3
3.4 Applications of Radian Measure
Recall: :
ϴ (in radians) =
s
,
r
Solve equation for ‘s’ to get equation for Arc Length on a Circle.
s=
Find the arc length, “s”
1) r = 3 m, θ = 315⁰
2) r = 5 cm
θ = 200⁰
Example 2 (pg 148) – For the ferris wheel, find the distance traveled by the rider for ϴ = 45o & ϴ = 105o
Definition: subtends - an angle subtended by an arc, line segment, or other curve is one
whose two rays pass through the endpoints of the arc
See Example 4 (pg 150): A person standing on earth notices that a 747 Jumbo Jet flying overhead subtends an
angle of 0.45o. If the length of the jet is 230 feet, find its altitude to the nearest thousand feet.
CA 3.4 #27, 29, 23, 35
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Math 170
Trigonometry Lecture Notes
Chapter 3
Area of a Sector (of a circle).
This is really just finding the area of a portion of a circle ( a slice of the pie)
Area of circle = πr2
Area of sector is a portion of the area of the circle
A=
θ
2
π r , ϴ in radians
2π
simplify…
Area sector =
*** Note you MUST be able to use these formulas ANY way. You MUST be ready to solve for whatever the
missing value is, not just the one that is currently solved for.***
1) If a 12” pizza is divided into 8 equal pieces, find the area of one piece. If the pizza is half an inch thick, find
the volume.
2) If a 10” pizza is divided into 6 equal pieces, find the area and volume of one piece (again assuming the pizza is
half an inch thick).
CA: 3.4 # 43, 49, 55
3.5 Linear & Angular Speed
Rotation of a Wheel: Look at how fast any point on the wheel is moving.
Linear velocity (speed), v = _____
Angular velocity (speed), ω = ____
How are linear speed & angular speed related?
If a point is moving with uniform circular motino on a cirlce of radius r, then
v = rω
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Math 170
Trigonometry Lecture Notes
Chapter 3
1) Let P be a point on a wheel with r = 10 cm, ω = π/8 rad/sec, t = 5 sec
a) Find v
b) find the distance traveled
c) angle “traveled”
In Class Activity Book Find the missing variables using Circular Motion:
a) θ = 4π/3, t = 2 sec, ω = __________
b) v = 3 ft/sec, r = 2 ft, ω = __________
c) s = 8π, r = 3 cm, t = 20 sec,
v = _________, θ = __________, ω = __________
d) ω = 3π/2 rad/sec; r = 7 in, t = 10 sec. v = _________, s = __________, θ = __________
Applications: (Do #3 on this page)
3) Clock Speeds:
 The second hand of a clock makes one rotation _________, in ___________ sec.
 The minute hand of a clock makes one rotation _________, in ___________ min or ________ sec.
 The hour hand of a clock makes one rotation _________, in ___________ hour or _________ min.
a) Find the linear and angular speed of the second hand of a clock, if the second hand is 5 in long.
b) Find the linear and angular speed of the minute hand of a clock, if the minute hand is 4 cm long. Express both
in units/minute and units/second.
CA 4) A wheel makes 36 rev/min. If the radius is 12 cm, find: v and ω
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