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Name:______________________________
NOTES 2.5, 6.1 – 6.3
Date:________________Period:_________
Mrs. Nguyen’s Initial:_________________
LESSON 2.5 – MODELING VARIATION
Direct Variation
y  mx  b when b  0 or
y  mx or y  kx
y  kx
-
Direct variation as
n
th
power
Inverse variation
and k  0 
y varies directly as x
y is directly proportional to x
k is the constant of variation
k is the constant of proportionality
Express the statement as an
equation. Use the given
information to find the constant of
proportionality.
B is directly proportional to L. If L
= 15, then B = 1350.
y  kx n and k  0 
- y varies directly as the nth power of x
- y is directly proportional to the nth power
of x
B is directly proportional to the
square of L. If L = 15, then B =
1350.
k

or k  xy
x
As x goes up, y goes down or
As y goes up, x goes down
n varies inversely as f. If n = 3,
then f = 5.
y
Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 1
Joint variation
z  kxy

z varies directly as x and y
M varies jointly as h and n. If h = 7
and n = 9, then M = 504.
Practice Problems:
1.
The pressure P of a sample of gas is directly
proportional to the temperature T and inversely
proportional to the volume V.
a.
Write an equation that expresses this variation.
b.
Find the constant of proportionality if 100L of
gas exerts a pressure 33.2 kPa at a temperature
of 400 K.
c.
If the temperature is increased to 500K and the
volume is decreased to 80L, what is the
pressure of the gas?
2.
The power P (measured in horse power, hp)
needed to propel a boat is directly proportional
to the cube of its speed s. An 80-hp engine is
needed to propel a certain boat at 10 knots.
Find the power needed to drive the boat at 15
knots.
Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 2
LESSON 6.1 – ANGLE MEASURE
Trigonometry
Measurement of triangles
Angles
An angle is in standard position if…
1. its initial side is along the positive x-axis
2. its vertex is at the origin, and
Note: Labeled with
Greek letters:
 ,  ,  ,...
Coterminal Angles
Angles with the same initial and terminal sides.
 and   360n
 and   2 n
Review
1. Central angle
2. Acute angle
3. Right angle
4. Obtuse angle
5. Arc measure
6. Arc length
7. Complementary angles
8. Supplementary angles
Practice Problems: The measure of an angle in standard position is given. Find two positive angles and two
negative angles that are co-terminal with the given angle. Sketch the angles.
1.
210
2.
45
Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 3
3.
540
Radian Measure
One radian
  is the measure of a central
Note: In a full
revolution, the arc
length s is equal to
C  2 r  s .
Also, there are just
over six radius lengths
in a full circle.
Therefore, the central
angle  that intercepts an arc s equal in
length to the radius of a circle.
C = 2r  6.28r
The radian measure of an angle of one full
revolution is 2 . Since one full circle has
360 ,
360  2
180  
rad
Degrees to Radians
Radians to Degrees
Multiply by
Multiply by
rad
90 

2

angle is 


s
r
where
is measured in
radians.
60 
rad

3
rad
Example:
180
180
Example:

Practice Problems: Convert the following angles from degrees to radians and from radians to degrees without
using a calculator.
4.
150 
5.
7

6
6.
240 
7.

11

30
Practice Problems: Convert the following angles from degrees to radians and from radians to degrees using a
calculator and round to 3 decimal places.
8.
87.4 
9.
2 
10.
0.54 
Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 4
11.
0.57 
Practice Problems: The measure of an angle in standard position is given. Find two positive angles and two
negative angles that are co-terminal with the given angle. Sketch the angles.
12.
13
6
13.
3
4
14.
In a circle of radius r, the length s of an arc that
subtends a central angle of  radians is: s  r
Length of a
Circular Arc
2
3
Example:
Note:  must be in radians.
Review Problem 15: Find the following arc lengths using geometry then use s  r to validate your
answers.
CB  24in ,
mA0 B  60 , find the
Given
1. Circumference =
following arc lengths using 2
methods
2. Length of
AB
3. Length of
=
CA
4. Length of
CDB
=
5. Length of
ADB
=
6. Length of
ADC
=
=
A
60
O
C
B
24 in
D
m
AB 

mCA



mCDB
m
ADB 

m
ADC 


Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 5
Practice Problems: Find the unknown value.
16.
A central angle  in a circle of radius 24 cm is
subtended by an arc of length 6 cm. Find the measure
of  in radians.
17.
18.
A bicycle’s wheels are 14 inches in diameter.
How far (in miles) will the bike travel if its wheels
revolve 500 times without slipping?
19.
How many revolutions will a Ferris wheel of
diameter 60 feet make as the Ferris wheel travels a
distance of a ½ mile?
20.
An ant is sitting 5 cm from the center of a c.d..
If the c.d. turns 40  , how far has the ant moved in
meters?
21.
A bug is on a car’s windshield wiper and is 10
inches from the base of the windshield wiper. If the
bug moves 34 inches, at what angle did the windshield
wiper turn?
Find the radius of the circle if an arc of length
8 in on the circle subtends a central angle of

4
.

The angular velocity of a point on a rotating object is the
number of degrees (radians, revolutions, etc.) per unit
time through which the point turns.

Linear Speed
The linear velocity of a point on a rotating object is the
distance per unit time that the point travels along its
circular path.
v
Note:
The linear velocity depends on how far the object is from the axis of rotation, whereas
the angular velocity is the same no matter where the object lies on the rotating object.
Angular Speed
t
s
t
Relationship
If a point moves along a circle of radius r with angular speed  , then its linear speed v
between Linear and is given by: v  r
Angular Speed
Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 6
Practice Problems: Solve the following problems.
22.
A woman is riding a bike whose wheels are 26
inches in diameter. If the wheels rotate at 125
revolutions per minute (rpm), find the speed at which
she is traveling, in miles per hour.
23.
The rear wheels of a tractor are 4 feet in
diameter, and turn at 20 rpm.
(a) How fast is the tractor going (feet per second)?
(b) The front wheels have a diameter of only 1.8 feet.
What is the linear velocity of a point on their tire
treads?
(c) What is the angular velocity of the front wheels in
rpm?
24.
The pedals on a bike turn the front sprocket at
8 radians per second. The sprocket has a diameter of
20 cm. The back sprocket, connected to the wheel, has
a diameter of 6 cm.
25.
Dan and Ella are riding on a Ferris wheel. Dan
observes that it takes 20 seconds to make
a complete revolution. Their seat is 25 feet from the
axle of the wheel.
(a) Find the linear velocity of the chain.
(a) What is their angular velocity?
(b) Find the angular velocity of the back sprocket.
(b) What is their linear velocity?
Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 7
LESSON 6.2 – TRIGONOMETRY OF RIGHT TRIANGLES
The Trigonometric
Ratios
Let  has an acute angle of a right triangle. The
six trigonometric functions of the angle  are
defined below.
sin  
opp
hyp
csc 
hyp
opp
cos 
adj
hyp
sec 
hyp
adj
tan  
opp
adj
cot  
adj
opp
Review: Special
Right Triangles
45
30
x 2
x
SOH CAH TOA
2x
x 3
45
90
x
90
60
x
Practice Review Problems: Evaluate the following
1.
a4
b
c
2.
a
b6 2
c
3.
a
b
c  10
4.
a7 3
b
c
5.
a
b  hat
c
6.
a
b
c  iPod 7
45
c
b
45
90
a
Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 8
7.
a4
b
c
8.
a
b6 2
c
9.
a
b
c  10
10.
a7 3
b
c
11.
a
b  hat
c
12.
a
b
c  iPod 7
30
b
c
90
60
a
Practice Problems: Evaluate the six trig functions at each real number without using a calculator.
1.
2.
sin  
csc 
17
4
cos 
sec 
tan  
cot  
sin  
csc 
cos 
sec  6
tan  
cot  
Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 9
Practice Problems: Evaluate without using a calculator. Draw and label triangles.
3.
tan 60 
4.
csc 45 
5.
tan 30 
6.
sec30 
Practice Problems: Evaluate the given expression without using a calculator. Leave your answer in simplest
radical form.
7.
sin 60  cos 30
8.
tan 45  cot(60)
9.
tan 60sec 60
10.
15sin 30 cos 45
Practice Problems: Solve the right triangle.
B
a
c
a
mA 
b
mB 
c
mC 
34.2
C
b=19.4
A
Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 10
12.
100
75
13.
72.3

6
Angles of Elevation
and Depression
a
mA 
b
mB 
c
mC 
a
mA 
b
mB 
c
mC 
The angle of elevation is the angle from a
horizontal line UP to an object.
The angle of depression is the angle from a
horizontal line DOWN to an object.
Practice Problems: Set up an equation for each word problem and solve.
14.
Suppose you have been assigned the job of
measuring the height of the local water tower.
Climbing makes you dizzy so you decide to do the
whole job at ground level. From a point 47.3 meters
from the base, you find that you must look up at an
angle of 53 degrees to see the top of the tower. How
high is the tower?
15.
When landing, a jet will average a 3 angle of
descent. What is the altitude, to the nearest foot, of a
jet on final descent as it passes over an airport radar 6
miles from the start of the runway?
16.
At a point 300 feet from the base of a building,
the angle of elevation to bottom of a smokestack
is 40 , and the angle of elevation to the top is 55 .
Find the height of the smokestack alone.
17.
The distance between a plane and a building on
the ground is 350 feet. The angle of depression from
the plane to the building is 20 . Find the horizontal
distance from the plane to the building.
Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 11
LESSON 6.3 – TRIGONOMETRIC FUNCTIONS OF ANGLES
Definitions of
Trigonometric
Functions of any
Angle
Let  be an angle in standard position with
a point on the terminal side of  and
r  x2  y 2  0 .
sin  
y
r
csc 
cos 
x
r
r
sec  , x  0
x
tan  
y
, x0
x
cot  
Practice Problem 1: Let  4,  3
be a point on the terminal side of
 . Find:
Practice Problem 2: Let
3
 1
 3 ,  7  be a point on the
4
 2
terminal side of  . Find:
Signs of the
Trigonometric
Functions
 x, y 
r
, y0
y
x
, y0
y
sin  
csc 
cos 
sec 
tan  
cot  
sin  
csc 
cos 
sec 
tan  
cot  
Use “All Student Take Calculus” to
figure out in which quadrant each trig
function has a positive value.
Students
All
sin/csc positive
Take
tan/cot positive
Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 12
Calculus
cos/sec positive
II
(-,+
)
sin :
cos :
tan :

csc :
sec :
sin :
cos :
tan :
csc :
sec :
III
(-,
Practice Problem 3: Find the
value of the six trigonometric
15
functions. Given: tan    ;
8
sin   0
Practice Problem 4: Find the
value of the six trigonometric
functions. Given: cot  is

3
undefined;   
2
2
cot :

I
2
(+ ,
sin :
cos :
tan :
+)
csc :
sec :
cot :
0
cot :
sin :
cos :
tan :
cot :
, -)
IV
3
)
csc :
sec :
(+
2
sin  
csc 
cos 
sec 
tan  
cot  
sin  
csc 
cos 
sec 
tan  
cot  
Practice Problem 5: Evaluate the following:
a.
sin 0 
c.
sin

2

b.
sin  
d.
sin
3

2
Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 13
Definition of
Reference Angle
Let  be an angle in standard position. Its reference angle is the acute angle  ' formed
by the terminal side of  and the x-axis.
Quadrant II
Quadrant III
Quadrant IV
 '      rad
 '      rad
 '   2    rad
 '  180    deg ree
 '    180  deg ree
 '   360    deg ree
Practice Problem 6: Find the reference angle  ' for the following. Graph the angle
a.
  309   ' 
c.

7
 ' 
4
Evaluating
Trigonometric
Functions of any
Angle
b.
  145   ' 
d.

11
 ' 
3
To find the value of a trig function of any  .
1. Determine the function value for the associated  ' .
2. Depending on the quadrant in which  lies, affix the appropriate sign to the
function value.
Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 14
FUNDAMENTAL TRIGONOMETRIC INDENTITIES
Reciprocal
Identities
sin  
1
csc 
cos  
1
sec 
tan  
1
cot 
csc  
1
sin 
sec  
1
cos 
cot  
1
tan 
Quotient
Identities
tan  
sin 
cos 
Pythagorean
Identities
sin 2   cos 2   1
cot  
cos 
sin 
1  tan 2   sec 2 
1  cot 2   csc 2 
Practice Problem 7: Evaluate the trig functions
a.
cos
4

3
c.
csc
11

4
Practice Problem 8:
Find the indicated
a.
trig function
cos  
5
8
b.
tan  210  
d.
cot

sin   
3
5
and Quad IV
cos 
Practice Problem 9: Find two exact solutions of the equation.
 0    2  .
cot   1
2
b.
and Quad III
sec 
a.

b.
places.

in degrees
 0    360  and radians
tan   2.3545 .
Mrs. Nguyen – Honors Algebra II – Chapter 2.5, 6.1 – 6.3 Notes – Page 15
Round to 2 decimal