Friday, August 31, 2012
Proofs of Heron's formula and the ½|ad - bc| area method
After class on Thursday, a student asked for proofs of our two methods for getting the area of a triangle.
Here is a link to the Wikipedia page on Heron's formula which has several proofs.
For the triangle, here is a page proving |ad - bc| is the area of a parallelogram, which is done by cutting a parallelogram out of a larger rectangle. Any parallelogram can thought of as two congruent triangles glued together along one corresponding side, so half the area of a parallelogram is the area of one of the triangles.
Saturday, August 25, 2012
Links to material covered in first week and on first homework.
The blog has entries from fall 2011, the last time I taught this course. I expect to follow much the same order of material this time around. Here are links to the topics we worked on this first week.
Classifications of triangles and The Triangle Inequality.
Practice on classification of triangles given two angles.
Practice on classification given three side lengths.
In class on Thursday, I asked for all possible integer length side triples for perimeter of 11 and perimeter of 13. Here are the answers.
11
5, 5, 1
5, 4, 2
5, 3, 3
4, 4, 3
13
6, 6, 1
6, 5, 2
6, 4, 3
5, 5, 3
5, 4, 4
Find both classifications for these triples, the classification by largest angle (obtuse, right, acute) and the classification by relations between side lengths (equilateral, isosceles, scalene).
Answers in the comments.
Tuesday, August 21, 2012
Syllabus for Fall 2012
Math 50: Trigonometry (L-41147) Fall 2012 – Laney College
Instructor: Matthew Hubbard TTh: 10:30-11:45 am
Email address: mhubbard@peralta.edu
Recommended Text: Trigonometry (open source) http://mecmath.net/trig/trigbook.pdf
Class website: http://thetrig.blogspot.com/
Office hours: MW: 8:30-8:55 am F-203 (Classroom)
T-Th: 10:00-10:25 am G-206 (Classroom)
Add and drop class dates
Last date to add: Sun., Sept. 2
Last date to drop class without a “W”: Sat., Sept. 1
Last date to drop class with a “W”: Sat., Nov. 17
Holiday schedule for T-Th schedule
Thanksgiving Thursday, Nov. 22
Test dates:
Midterm 1: Thursday, Sept. 27
Midterm 2: Thursday, Nov. 1
Comprehensive Final: Thursday, Dec. 13 10:00 am- noon
Homework to be turned in: Assigned last class period of the week, due next class.
Late homework accepted AT THE BEGINNING of the class after it was due
Quizzes: One on the last class of the week in weeks without a midterm or final
Grading system
Homework 20%
Lab 5%
Quizzes 25%
Midterm I 25%
Midterm II 25%
Final 25%
Voting bonus 2%
If you are eligible to vote, bring an “I voted” sticker
If you are too young to vote, bring proof of age
If you are not a U.S. citizen, bring proof of status
Lowest two of the homework scores will be dropped from the total.
Lowest two of the quiz scores will be dropped from the total.
Lowest total out of 100 points the quiz total and two midterms will be dropped from the final grade.
Anyone getting a higher grade out of 100 points on the final than the weighted average of all grades combined will get the final percentage instead deciding the final grade. This option is only available to students who have missed at most two homework assignments.
Anyone going into final with 97.5% class average can skip the final and be awarded an A.
Class rules: All cell phones and electronic communication devices off during class.
No hats, hoodies or headphones worn during quizzes and exams.
No calculators that also combine a cell phone or text message machine.
Recommended calculator: TI-30XIIs (any calculator with at least two lines of output will do, the TI-30XIIs is the cheapest that does all the things you need to do in this class. If you need help with any Texas Instruments calculator, I should be able to steer you in the right direction. I haven’t used other brands of calculators as much.)
Open source textbook: The textbook is free and online. It helps a lot if you have a computer and Internet at home so you can have your own copy on your computer. You will need the Adobe Reader software, which is free and can be downloaded from several websites. I will be assigning problems from the book occasionally.
Academic honesty: All assignments you turn in, homework, exams and quizzes, must be your own work. Anyone caught cheating on these assignments will be punished, where the punishment can be as severe as failing the class or being put on college wide academic probation.
Students with disabilities
The Disabled Students Program Services (DSPS) should have your academic accommodation with the instructor. After the first day, I will accept these accommodations electronically or by hard copy on paper. If you need academic accommodation and have not yet applied, please call 510-464-3428 for an appointment.
Exam policies
Tests will be closed book and closed notes. Some information you will be expected to remember, other formulas and information will be provided. No sharing of calculators is allowed. You are responsible for knowing how to use your calculator to find answers.
Student learning outcomes
Math 50 Trigonometry
• Evaluate the 6 trigonometric functions using a calculator, as well as determining exact values for some special angles without a calculator.
• Solve a triangle (right, acute, obtuse), given various angles and sides.
• Convert between decimal degrees, degree-minute-seconds, and radian measure of an angle.
• Demonstrate knowledge of several trigonometric identities and use them to verify other identities.
• Graph trigonometric functions.
• Solve trigonometric equations.
The reciprocal relationship
The teacher will be on time and prepared to teach the class.
The students will be on time and prepared to learn.
The teacher will present the material to the best of his ability.
The students will absorb the material to the best of their ability. They will ask questions when topics are not clear.
The teacher will do his best to answer the questions the students ask about the material, either by repeating an answer with more details included or by taking a different approach to the material that might be clearer to some students.
The students will understand if the teacher feels a topic has been covered enough for the majority of the class and will accept questions being answered outside the class, either in extra time or through written communication.
The teacher will do his best to keep the class about the material. Personal details and distractions that are not germane to the class should not be part of the class.
The students will do their best to keep the class about the material. Questions that are not about the topic should be avoided. Distractions like cell phones and texting are not welcome when the class is in session.
The teacher will give assignments that will help the students master the skills required to pass the course.
The students will put in their best efforts to complete the assignments.
When the assignments are completed, the teacher will make every effort to get the assignments graded and back to the students in a timely manner, by the next class session whenever possible.
The teacher will present real life situations where the skills being learned will be used when they exist. In math, sometimes a particular skill is needed in general to solve later problems that will have real life applications. Other skills have the application of “learning how to learn”, of committing an idea to memory so that committing other ideas to memory becomes easier in the long run.
The student has the right to ask “When will I use this?” when dealing with mathematical topics. Sometimes, the answer is “We need this skill for the next skill we will learn.” Other times, the answer is “We are learning how to learn.” Both of these answers are as valid in their way as “We will need this to understand perspective” or “We use this to balance our checkbooks” or “Ratios can be used to figure out costs” or other real life applications.
Instructor: Matthew Hubbard TTh: 10:30-11:45 am
Email address: mhubbard@peralta.edu
Recommended Text: Trigonometry (open source) http://mecmath.net/trig/trigbook.pdf
Class website: http://thetrig.blogspot.com/
Office hours: MW: 8:30-8:55 am F-203 (Classroom)
T-Th: 10:00-10:25 am G-206 (Classroom)
Add and drop class dates
Last date to add: Sun., Sept. 2
Last date to drop class without a “W”: Sat., Sept. 1
Last date to drop class with a “W”: Sat., Nov. 17
Holiday schedule for T-Th schedule
Thanksgiving Thursday, Nov. 22
Test dates:
Midterm 1: Thursday, Sept. 27
Midterm 2: Thursday, Nov. 1
Comprehensive Final: Thursday, Dec. 13 10:00 am- noon
Homework to be turned in: Assigned last class period of the week, due next class.
Late homework accepted AT THE BEGINNING of the class after it was due
Quizzes: One on the last class of the week in weeks without a midterm or final
Grading system
Homework 20%
Lab 5%
Quizzes 25%
Midterm I 25%
Midterm II 25%
Final 25%
Voting bonus 2%
If you are eligible to vote, bring an “I voted” sticker
If you are too young to vote, bring proof of age
If you are not a U.S. citizen, bring proof of status
Lowest two of the homework scores will be dropped from the total.
Lowest two of the quiz scores will be dropped from the total.
Lowest total out of 100 points the quiz total and two midterms will be dropped from the final grade.
Anyone getting a higher grade out of 100 points on the final than the weighted average of all grades combined will get the final percentage instead deciding the final grade. This option is only available to students who have missed at most two homework assignments.
Anyone going into final with 97.5% class average can skip the final and be awarded an A.
Class rules: All cell phones and electronic communication devices off during class.
No hats, hoodies or headphones worn during quizzes and exams.
No calculators that also combine a cell phone or text message machine.
Recommended calculator: TI-30XIIs (any calculator with at least two lines of output will do, the TI-30XIIs is the cheapest that does all the things you need to do in this class. If you need help with any Texas Instruments calculator, I should be able to steer you in the right direction. I haven’t used other brands of calculators as much.)
Open source textbook: The textbook is free and online. It helps a lot if you have a computer and Internet at home so you can have your own copy on your computer. You will need the Adobe Reader software, which is free and can be downloaded from several websites. I will be assigning problems from the book occasionally.
Academic honesty: All assignments you turn in, homework, exams and quizzes, must be your own work. Anyone caught cheating on these assignments will be punished, where the punishment can be as severe as failing the class or being put on college wide academic probation.
Students with disabilities
The Disabled Students Program Services (DSPS) should have your academic accommodation with the instructor. After the first day, I will accept these accommodations electronically or by hard copy on paper. If you need academic accommodation and have not yet applied, please call 510-464-3428 for an appointment.
Exam policies
Tests will be closed book and closed notes. Some information you will be expected to remember, other formulas and information will be provided. No sharing of calculators is allowed. You are responsible for knowing how to use your calculator to find answers.
Student learning outcomes
Math 50 Trigonometry
• Evaluate the 6 trigonometric functions using a calculator, as well as determining exact values for some special angles without a calculator.
• Solve a triangle (right, acute, obtuse), given various angles and sides.
• Convert between decimal degrees, degree-minute-seconds, and radian measure of an angle.
• Demonstrate knowledge of several trigonometric identities and use them to verify other identities.
• Graph trigonometric functions.
• Solve trigonometric equations.
The reciprocal relationship
The teacher will be on time and prepared to teach the class.
The students will be on time and prepared to learn.
The teacher will present the material to the best of his ability.
The students will absorb the material to the best of their ability. They will ask questions when topics are not clear.
The teacher will do his best to answer the questions the students ask about the material, either by repeating an answer with more details included or by taking a different approach to the material that might be clearer to some students.
The students will understand if the teacher feels a topic has been covered enough for the majority of the class and will accept questions being answered outside the class, either in extra time or through written communication.
The teacher will do his best to keep the class about the material. Personal details and distractions that are not germane to the class should not be part of the class.
The students will do their best to keep the class about the material. Questions that are not about the topic should be avoided. Distractions like cell phones and texting are not welcome when the class is in session.
The teacher will give assignments that will help the students master the skills required to pass the course.
The students will put in their best efforts to complete the assignments.
When the assignments are completed, the teacher will make every effort to get the assignments graded and back to the students in a timely manner, by the next class session whenever possible.
The teacher will present real life situations where the skills being learned will be used when they exist. In math, sometimes a particular skill is needed in general to solve later problems that will have real life applications. Other skills have the application of “learning how to learn”, of committing an idea to memory so that committing other ideas to memory becomes easier in the long run.
The student has the right to ask “When will I use this?” when dealing with mathematical topics. Sometimes, the answer is “We need this skill for the next skill we will learn.” Other times, the answer is “We are learning how to learn.” Both of these answers are as valid in their way as “We will need this to understand perspective” or “We use this to balance our checkbooks” or “Ratios can be used to figure out costs” or other real life applications.
Sunday, December 11, 2011
Example similar to the take home final
By request, I present an example like the take home final. This is the graph of r =2cos(2theta). I calculated the values at all 24 points that have lines radiating from the origin and marked them, then I played connect the dots, smoothing the curve instead of drawing straight lines. Here are the values at all 24 points.
r = 2cos(2*0) = 2
r = 2cos(2*pi/12) = sqrt(3)
r = 2cos(2*pi/6) = 1
r = 2cos(2*pi/4) = 0
r = 2cos(2*pi/3) =-1
r = 2cos(2*5pi/12) =-sqrt(3)
r = 2cos(2*pi/2) =-2
r = 2cos(2*7pi/12) =-sqrt(3)
r = 2cos(2*2pi/3) =-1
r = 2cos(2*3pi/4) = 0
r = 2cos(2*5pi/6) = 1
r = 2cos(2*11pi/12)
= sqrt(3)
r = 2cos(pi) = 2
r = 2cos(2*13pi/12)
= sqrt(3)
r = 2cos(2*7pi/6) = 1
r = 2cos(2*5pi/4) = 0
r = 2cos(2*4pi/3) =-1
r = 2cos(2*17pi/12)
=-sqrt(3)
r = 2cos(2*3pi/2) =-2
r = 2cos(2*19pi/12)
=-sqrt(3)
r = 2cos(2*5pi/3) =-1
r = 2cos(2*7pi/4) = 0
r = 2cos(2*11pi/6) = 1
r = 2cos(2*23pi/12)
= sqrt(3)
The result is called a four leaf rose.With the problem from the take home, because we have the formula r = 2 + cos(3*theta), the radius will never get to be less than zero, so it will not cross the origin mulitple times like this one does.
Friday, December 9, 2011
The take home section of the final
Here is the take-home section of the final as a picture file. You should be able to download it, but if not, send me an e-mail at mhubbard(at)peralta(dot)com, substituting the correct symbols for the words in parentheses.
I can also send you a list of the topics for the final if you did not get it in class today. You are allowed to have notes on two 3" x 5" cards.
See you on Wednesday, Dec. 14 at 8:00 am, one hour earlier than usual.
Monday, November 28, 2011
Polar coordinates.
In class Monday, we introduced the idea of polar coordinates. In rectilinear coordinates (x, y), the first number tells us how far right (positive) or left (negative) we move on the horizontal axis and the second number tells us how far up (positive) or down (negative) we move on the vertical axis.
In polar coordinates we have (r, theta). There is a central point, not unlike (0, 0) in the xy-axis system, and we consider that we are pointing in a direction we call angle 0 in radians. The standard is to make that direction to the right, the same as the positive x-axis in rectilinear. r is the distance from the origin and theta is the angle given in radians.
It is acceptable to have negative values for distance and for angles.
Unlike rectilinear coordinates, polar coordinates are not unique. The easiest example of this are the coordinate pairs (0, 0) and (0, 1). The first zero means we are a distance of 0 from the origin. That means we are on the origin. The angle we turn doesn't change where we are.
Even when points aren't the origin, the polar coordinates are not unique. The simplest example is
(1, 0) = (1, 2pi)
What this says is if we are 1 away from the origin and pointing to the right, this is the same as being 1 away from the origin and turning a full circle. In fact, adding or subtracting any multiple of 2pi to an angle brings us back to pointing in the exact same direction.
We will be working more in polar coordinates for the rest of the week, possibly longer.
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Monday, November 21, 2011
step by step through some half angles of famous angles.
The half angle formula deal with fractions and square roots. The Blogger text software doesn't have good ways to type those in, so I resort to using pictures with captions.
These are the original half angle formulas.
This is a step by step (across) way to turn the sine and cosine of 15 degrees into fractions with whole number denominators. These answers look different from the ones you get using subtraction of angles, either 60 degrees - 45 degrees or 45 degrees - 30 degrees. If you type in the formulas, you get the same answers on your calculators.
This is a step by step (across) way to turn the sine and cosine of 22.5 degrees into fractions with whole number denominators.
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