Transcript Document

Welcome to the Unit 1 seminar of Intro to Linear Algebra!
My name is Cathy Johnson. I am here to help you the next 10
weeks! I am looking forward to this term and hoping you
are too.
I’d like to begin by reviewing a few items from the syllabus,
going over ground rules for seminar, and seeing if you have
any immediate questions for me.
Everyone: Questions before we begin?
Syllabus: please read the syllabus which is the same whether
you downloaded it from Doc Sharing, read it under Course
Home: Syllabus, or read the one I emailed you on the first
day of class. Key things include attendance requirements,
due dates and late policies, showing your work in your
Project assignments, and doing your own work. There have
been problems with plagiarism in recent quarters, so all
instructors are on the lookout, and the consequences are
harsh.
How do you contact me?
My AIM id is cjohnsonkap. If I am logged into AIM, you are
always welcome to send me questions and I will answer if
possible. Whenever you AIM me, make sure to identify
yourself. The username "bearsfan123" doesn't tell me who
you are. If there is a specific time you would like me to
meet you online please email me and I will do my best to
help you.
Email address: My Kaplan email address is
”[email protected]". Make sure my address is
added to your address book so that your spam blocker
doesn't accidentally block my emails to you.
Seminar: Held Wednesdays 8-9 PM ET
Due dates and late assignments: All projects, discussion
question posts, and seminar option II, are due at 11:59
pm ET the Tuesday that concludes each Unit. All Unit 1
assignments are due this Tuesday. Late projects and
seminar option II assignments will be accepted, but
penalized 2% per day late. Late message board posts
will be scored down 10% per week late. Late projects
and discussion board posts will not be accepted after
11:59 pm ET Sunday of unit 10.
Extraordinary Circumstances: If you have extraordinary
circumstances that prevent you from participating in
class, you need to contact me. Good communication is
the key to overcoming obstacles you may encounter
during the term. I am more than willing to work with you,
but I have to know about the situation to do that.
I also want to mention that I will post my notes and
PowerPoint to the doc sharing folder each week. In
addition, you can print the “history” of the seminar so there
is no need to take complete notes.
Everyone: Questions?
Ok, let’s get started on the material. As we discuss the
material, please feel free to let me know if you have any
questions.
This first unit deals with
• angles and their measure,
• conversion of angle measure from radians to/from degrees,
• trigonometric functions, and
• right triangle trigonometry.
Basic Definitions
• A ray is a half-line that begins at a certain point and
extends forever in one direction.
• The point at which the ray begins is called the endpoint.
• When a ray is rotated about its fixed end point, it moves
from it initial position to it final position. As it does this it
creates an angle between the two positions.
• Angles can be measured in degrees. A complete
revolution of the ray from its initial position back to its
initial position is 360 degrees.
• A degree can be subdivided and can be written as either a
decimal or a fraction. For example, ½ degree or 0.5
degree.
Another system of subdivision is called the sexagesimal
system. In that system 1 degree is equal to 60 minutes
and 1 minute is equal to 60 seconds.
Symbolically, this is written as
1° = 60’ and 1’ = 60”
To convert between the a decimal (or fractional) part of a
degree to this system, you would use the following
conversions.
1° = 60’ = 3600”
Everyone: Questions?
Converting Angle Decimal and Sexagesimal Measures
To convert, use the following:
1° = 60’ = 3600”
Example:
Solution:
Convert 28.6 degrees to the sexagesimal system.
28.6 degrees = 28 ° 0.6(60)’
= 28 ° 36’
Everyone: Questions?
Everyone: Convert 123.4 degrees to the sexagesimal system.
Converting Angle Decimal and Sexagesimal Measures
To convert, use the following:
1° = 60’ = 3600”
Example:
Solution:
Convert 28.6 degrees to the sexagesimal system.
28.6 degrees = 28 ° 0.6(60)’
= 28 ° 36’
Everyone: Questions?
Everyone: Convert 123.4 degrees to the sexagesimal system.
Solution:
123.4 degrees = 123 ° 0.4(60)’
= 123 ° 24’
Example: Convert 42 ° 36’ 41” to degrees
42 ° 36’ 41”
= (42 + 36/60 + 41/3600) °
= 42.6114 °
Everyone: Questions?
Everyone: Convert 21 ° 14’ 20” to degrees
Example: Convert 42 ° 36’ 41” to degrees
42 ° 36’ 41”
= (42 + 36/60 + 41/3600) °
= 42.6114 °
Everyone: Questions?
Everyone: Convert 21 ° 14’ 20” to degrees
21 ° 14’ 20”
= (21 + 14/60 + 20/3600) °
= 21.2389 °
Everyone: Questions?
One other unit for measuring angles is called the radian. If
you let the fixed point of the array be the center of a circle,
then the angle that is created when the arc length is equal
to the radius length is equal to 1 radian. The angle created
in one full revolution of the ray is equal to 2π radians.
1 revolution = 2π radians = 360 °
1 rad
same length
Converting Degree and Radian Angle Measures
To convert between degrees and radians set up a proportion
using the following:
1 revolution = 2π radians = 360 °
Example:
Solution:
Convert 30 degrees to radians
30 degrees
=
x radians
360 degrees
2π radians
Solve:
x radians
= (30 degrees/360 degrees) 2π radians
= (1/12) 2π radians
= π / 6 radians
Everyone: Questions?
Everyone: Convert 75 degrees to radians
Everyone: Convert 75 degrees to radians
Solution:
75 degrees =
x radians
360 degrees
2π radians
Solve:
x radians = (75 degrees/360 degrees) 2π radians
= (5/24) 2π radians
= 10π / 6 radians
Everyone: Questions?
Everyone: Convert 1.47 radians to degrees
Everyone: Convert 1.47 radians to degrees
Solution:
x degrees
1.47 radians =
2 π radians
x degrees
360 degrees
= ( 1.47 radians / 2 π radians) 360 degrees
= 84.2 degrees
Everyone: Questions?
Trigonometric Functions
r
a
y
x
sine α = sin α = y/r = opposite side / hypotenuse
cosine α = cos α = x/r = adjacent / hypotenuse
tangent α = tan α = y/x = opposite / adjacent
cotangent α = cot α = x/y = adjacent / opposite
secant α = sec α = r/x = hypotenuse / adjacent
cosecant α = csc α = r/y = hypotenuse / opposite
Pythagorean Theorem
r2 = x2 + y2
α
r
y
x
Example: Find the values of the 6 trig functions for α.
α
r
7.2
3.4
First, determine r.
r2
= (7.2)2 + (3.4)2
= 51.84 + 11.56
= 63.4
r
= 7.96
Example: Determine the value of the six trigonometric
functions for the angle α.
r = 7.96
a
sin α
cos α
tan α
cot α
sec α
csc α
y = 7.2
x = 3.4
=
=
=
=
=
=
y/r
x/r
y/x
x/y
r/x
r/y
=
=
=
=
=
=
7.2 / 7.96
3.4 / 7.96
7.2 / 3.4
3.4 / 7.2
7.96 / 3.4
7.96 / 7.2
Everyone: Questions?
=
=
=
=
=
=
0.905
0.427
2.118
0.472
2.341
1.106
Everyone: Determine the value of the six trigonometric
functions for the angle α.
r
a
6
4
Everyone: Determine the value of the six trigonometric
functions for the angle α.
r
a
6
4
r = sqrt( 62 + 42
sin α = y/r =
cos α = x/r =
tan α = y/x =
cot α = x/y =
sec α = r/x =
csc α = r/y =
) = 7.21
6 / 7.21
4 / 7.21
6/4
4/6
7.21 / 4
7.21 / 6
Everyone: Questions?
=
=
=
=
=
=
0.832
0.555
1.5
0.667
1.803
1.202
Evaluating Trigonometric Functions
Okay, let’s use a calculator to evaluate trigonometric
functions when the angle is known and measured in both
degrees and radians.
make sure your calculator is on degree mode
cos (23 °) = 0.921
make sure your calculator is on radian mode
cos (4.17 rad) = -0.516
Everyone: Questions?
The procedure for the sine and tangent functions would be
similar. However, the remaining three trigonometric
functions are not as direct. For those, you need the
following reciprocal identities.
cot α =
1 / tan α
tan α = 1 / cot α
sec α = 1 / cos α
cos α = 1 / sec α
csc α = 1 / sin α
sin a = 1 / csc α
Let’s try a few examples.
sec (23 °) = 1 / cos (23 °) = 1 / 0.921 = 1.086
sec (4.17 rad) = 1 /(-0.516) = -1.937
You can use the basic trig definitions and the Pythagorean
theorem to solve right triangles. Let’s continue with an
earlier problem and show how to identify all sides and
angles of the following right triangle.
r = 7.96
y = 7.2
α
x = 3.4
To find the value of the angle α, you’ll use the inverse cosine
function on your calculator. It is indicate as follows:
cos-1x
Everyone: Questions?
r = 7.96
7.2
α
3.4
Now, let’s use the value for cos α that we found earlier, 0.427.
cos-1(0.427) = 1.13 rad = 64.72 °
Since the sum of the interior angles of a triangle is 180
degrees, the remaining angle will be
90° – 64.72° = 25.28 °
Everyone: Questions?
Everyone: Solve the following right triangle. (all sides, all
angles)
r
β
α
4.8
5.1
Everyone: Solve the following right triangle. (all sides, all
angles)
r
β
5.1
α
4.8
r = sqrt(5.12 + 4.82) = 7.004
cos α = 4.8/7.004 = 0.6853
cos-1 (0.6853) = 46.74 °
Β = (90 – 46.74) ° = 43.26 °
Everyone: Questions?
Applications
Right triangles are often used to find heights and distances of
objects. To do so, sketch a right triangle, label the parts of the
triangle that are given, and solve for the remaining parts that are
unknown.
Definition: Angle of Elevation
The angle that the line of sight from the top of an object makes with
the horizontal is called its elevation.
Definition: Angle of Depression
When an object from a lower level is viewed from a vantage point,
the angles that the line of sight makes with the horizontal is called
the angle of depression.
Please look at Example 17 in the handout given in the course shell.
Everyone: Questions?