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Math 1330 ONLINE
Week 5 Session Notes
Instructor: Dianne L. Gross
Email: [email protected]
Course Website: online.math.uh.edu/courses. Click on the link to Math 1330.
CASA Website: casa.uh.edu
Ready for quiz 5 and 6 after this week’s lecture.
Open Popper 5b
Review of last week and additional Systems problems:
Example 1: Find the coordinates of the center and radius for the given circle.
+
+ 4 − 6 − 23 = 0
Example 2: State the vertices eccentricity and equations of the asymptotes for the following;
9
−
16
=1
Last week we worked on solving systems of non-linear equations. For some it was very
abstract so here are the solutions drawn. Some pictures to see if that would help. Then we
will solve some more.
From week 4:
Example 16:
f ( x) = −2 x 2 + 8 x − 5
g ( x) = 6 x − 5
Solutions: 0, 5 and 1,1
Example 17:
x2 + y2 = 4
4x 2 − y 2 = 1
Solutions: 1, √3 , 1, √3 ,
1, √3 ,
1, √3
Example: 3: Solve the system of equations:
+
=
Possible Solutions:
Example 4: Solve the system of equations: 4
4
+
−
=9
+3
=4
=4
−
Popper 1: State the equations the asymptotes for the following:
A.
=−
D. "
= −$ ,
, =
! B. " = −
=$ #
,
=
#
=1
C.
=−
$
, =
$
!
Section 4.1
Special Triangles and Trigonometric Ratios
In this section, we’ll work with some special triangles before moving on to defining the six trigonometric
functions.
Two special triangles 30° − 60° − 90° and 45° − 45° − 90° triangles. With additional information, you should
be able to find the lengths of all sides of one of these special triangles.
Important Triangles
30-60-90 triangles
60 o
2x
x
30 o
x 3
Example 5: Find x and y if AC = 4√2.
Example 6: In the figure below, an altitude is drawn to the base of equilateral triangle ABC. If
AC = 8, find a, b and c.
45-45-90 triangles
45 o
x 2
x
45 o
x
Example 7:
A. Find x if the side AC = 5√3.
B.
.
The Six Trigonometric Ratios of an Angle
The word trigonometry comes from two Greek roots, trignon, meaning “having three sides,” and meter,
meaning “measure.” We have already defined the six basic trigonometric functions in terms of a right triangle
and the measure of three sides.
A trigonometric function is a ratio of the lengths of the sides of a triangle. If we fix an angle, then as to that
angle, there are three sides, the adjacent side, the opposite side, and the hypotenuse. We have six different
combinations of these three sides, so there are a total of six trigonometric functions. The inputs for the
trigonometric functions are angles and the outputs are real numbers.
The names of the six trigonometric functions, along with their abbreviations, are as follows:
Name of Function
cosine
sine
tangent
secant
cosecant
cotangent
Abbreviation
cos
sin
tan
sec
csc
cot
Let θ be an acute angle places in a right triangle; then
Side
opposite to
angle θ
Hypotenuse
Side adjacent to angle θ
θ
For ease of memorization
length of side adjacent to angle θ
length of hypotenuse
length of side opposite to angle θ
sin θ =
length of hypotenuse
length of side opposite to angle θ
tan θ =
length of side adjacent to angle θ
adjacent
hypotenuse
opposite
sin θ =
hypotenuse
opposite
tan θ =
adjacent
length of hypotenuse
sec θ =
length of side adjacent to angle θ
hypotenuse
sec θ =
adjacent
cos θ =
cos θ =
A useful mnemonic
device:
SOH-CAH-TOA
csc θ =
length of hypotenuse
length of side opposite to angle θ
csc θ =
hypotenuse
opposite
cot θ =
length of side adjacent to angle θ
length of side opposite to angle θ
cot θ =
adjacent
opposite
Note: For acute angles the values of the trigonometric functions are always positive since they are ratios of
lengths.
Example 8: Find the values of all six trigonometric ratios for the angle θ in the figure below.
θ
6
3
Example 9: Suppose a triangle ABC has C = 90o, AC = 7 and AB = 9. Find csc(A) and tan(B).
Use the following function for poppers questions 2 and 3.
3
+
+1
=4
Popper 2: State the center of the above function.
A. −3,1 B. 3, −1
C. 1, 3
D. −1, 3
Popper 3: State the length of the major axis.
A. 4
B. 8
C. 16
D. 2
Example 10: Suppose that %is an acute angle in a right triangle and secθ =
*√+
. Find sin% and cos %.
Section 4.2
Radians, Arc Length, and Area of a Sector
An angle is formed by two rays that have a common endpoint (vertex). One ray is the initial side and the other
is the terminal side. We typically will draw angles in the coordinate plane with the initial side along the
positive x axis.
A
Terminal side
θ
Vertex B
Initial Side
C
∠B, ∠ABC , ∠CBA, and θ are all notations for this angle. When using the notation ∠ABC and ∠CBA, the vertex
is always the middle letter. We measure angles in two different ways, both of which rely on the idea of a
complete revolution in a circle. The first is degree measure. In this system of angle measure one complete
1
revolution is 360° . So one degree is
of the circle.
360
The second method is called radian measure. One complete revolution is 2π . The problems in this section are
worked in radians. Radians are a unit free measurement. Suppose I draw a circle from the center and construct
an angle by drawing rays from the center of the circle to two different points on the circle in such a way that the
length of the arc intercepted by the two rays is the same as the radius of the circle. The measure of the central
angle thus formed is one radian.
arc length = r
r
θ =1
r
The Radian Measure of an Angle
Place the vertex of the angle at the center of a circle of radius r. Let s denote the length of the arc intercepted by
s
the angle. The radian measure θ of the angle is the ratio of the arc length s to the radius r. In symbols, θ = .
r
In this definition it is assumed that s and r have the same linear units. You can also solve the previous formula
in the form - = .%
One radian measure is the measure of the central angle (vertex of the angle is at the center of the circle) of a
circle that intercepts an arc equal in length to the radius of the circle.
If an angle has a measure of 2.5 radians, we write θ = 2.5 radians or θ = 2.5. There should be no confusion as
to whether radian or degree measure is being used. If θ has a degree measure of, say, 2.5 we must write θ =
2.5 o and not θ = 2.5.
Example 11: A circle has radius12 inches. A central angle % intercepts an arc of length 36 inches. What is the
radian measure of the central angle?
Example 12: A central angle, θ =
π
2
, in a circle intercepts an arc of length
12π
m. What is the radius of the
5
circle?
Use the following function for the popper questions 4 and 5:
+
*
=1
Popper 4: State the coordinates of the vertices.
A.
2,0 /01 2,0
B.
5,0 /01 5,0
Popper 5: State the length of the major axis.
A. 25
B. 4
C. 10
D2
C. 0, 2 /01 0, 2
D. 0, 5 /01 0, 5
Relationship between Degrees and Radius
How can we obtain a relationship between degrees and radians? We compare the number of degrees and the
number of radians in one complete rotation in a circle. We know that 360 o is all the way around a circle. The
length of the intercepted arc is equal to the circumference of the circle. Therefore, the radian measure of this
central angle is the circumference of the circle divided by the circle’s radius, r. The circumference of a circle of
a radius r is 2πr .
We use the formula for radian measure to find the radian measure of the 360 o angle. We know that the
circumference of a circle is 22.. So in this case, - = 22.. So the radian measure of a central angle in the case
of a complete a complete revolution:
- 22.
= 22
%= =
.
.
So, 360 o = 2π radians.
180° = 2./15/06
90° = ./15/0-
Dividing both sides by 2, we get 180 o = π radians. Dividing this last equation by 180 o or π gives the conversion
rules that follow:
Conversion between Degrees and Radians
Using the fact that π radians = 180 o ,
1. To convert degrees to radians, multiply degrees by
π radians
.
180 o
180 o
.
2. To convert radians to degrees, multiply radians by
π radians
Note: The unit you are converting to appears in the numerator of the conversion factor.
Example 13: Convert each angle in degrees to radians.
a. 150 o
b. - 135 o
Example 14: Convert each angle in radians to degrees.
π
a.
radians
3
b. −
11π
radians
6
2
Popper 6: Describe the graph of this equation.
A. Point
B. Hyperbola
7
= 0.
=1
C. Parabola
Common Angles (Memorize these!)
360 o = 2 π
60 o =
+2
B. one line C. Two intersection lines D. There is no graph
Popper 7: Classify the following:
A. Ellipse
+
π
3
180 o = π
45 o =
π
4
90 o =
30 o =
D. Circle
π
2
π
6
Sector Area Formula
In a circle of radius r, the area A of a sector with central angle of radian measure θ is given by A =
1 2
r θ.
2
Example15: Given a circle the area of sector is
π 2
π
in and the central angle is . Find the radius
3
6
Example 16: Find the perimeter of a sector with central angle 60° and radius 3 m.
To find the area of a sector of a circle, think of the sector as simply a fraction of the circle. If the central angle %
defining the sector is given in degrees, then we can use the following formula:
Example 17: Use the formula above to find the area of a sector, where % = 315° and r = 4 cm.
Use the following for popper questions 8 and 9:
=
Popper 8: The graph opens?
A. Left
B. Right
C. Up
D. Down
Popper 9: State the focal point.
A.
,8
B.
,8
C. 8,
D. 8,
$
Linear and Angular Velocity (Speed)
Consider a merry-go-round
edupics.com
The ride travels in a circular motion.
Some of the horses are right along the edge of the merry-go-round, and some are closer to the center. If you are
on one of the horses at the edge, you will travel farther than someone who is on a horse near the center. But the
length of time that both people will be on the ride is the same. If you were on the edge, not only did you travel
farther, you also traveled faster.
However, everyone on the merry-go-round travels through the same number of degrees (or radians).
There are two quantities we can measure from this, linear velocity and angular velocity.
The linear velocity of a point on the rotating object is the distance per unit of time that the point travels along its
circular path. This distance will depend on how far the point is from the axis of rotation (for example, the
center of the merry-go-round).
We denote linear velocity by v. Using the definition above, v =
s
, where s is the arclength ( s = rθ .)
t
The angular velocity of a point on a rotating object is the number of degrees (or radians or revolutions) per unit
of time through with the point turns. This will be the same for all points on the rotating object.
We let the Greek letter ω (omega) represent angular velocity. Using the definition above, ω =
θ
t
We can establish a relationship between the two kinds of speed by dividing both sides of the arc length formula,
s = rθ , by t.
s rθ
θ
=
=r
t
t
t
Linear Speed in Terms of Angular Speed
The linear velocity, v, of a point a distance r from the center of rotation is given by v = rω , where ω is the
angular velocity in radians per unit of time.
Example 18: If the speed of a revolving gear is 25 rpm (revolutions per minute),
a. Find the number of degrees per minute through which the gear turns.
b. Find the number of radians per minute through which the gear turns.
Example 19: A car has wheels with a 10 inch radius. If each wheel’s rate of turn is 4 revolutions per second,
a. Find the angular speed in units of radians/second.
b. How fast (linear speed) is the car moving in units of inches/second?
Example 20: Back in the day, we used to play record albums that rotated about the turntable
at 33 1/3 revolutions per minute. This was before there were CDs. How many radians per
minute is this? Find the linear velocities at the needle when it is just starting the record,
about 14.5 centimeters from the center
Popper 10: Find the coordinates of the center and state the radius for the given circle.
+
A.
,8
+
= $9
and the radius was 7