Two different ways to get the area of a triangle.

If we have three lengths, let's call them u, v and w, we have ways to find out if they could be the three sides of a triangle and if so, we can classify the triangle as isosceles, equilateral or scalene, as well as telling if the triangle is obtuse, right or acute.  With Heron's Formula (sometimes known as Hero's Formula), we can find the area.

Let p be the perimeter, u+v+w.  We will call the semi-perimeter s = ½(u+v+w). The formula for the area is sqrt(s(s-u)(s-v)(s-w)).  The simplest way to do this is to find the semi-perimeter and plug numbers in.

Example #1:  Let the sides be 5, 6 and 7.  The perimeter is 18, so s is 9.  The area is as follows.

sqrt(9(9-5)(9-6)(9-7)) =
sqrt(9 × 4 × 3 × 2) =
6 × sqrt(6) (square units)

Example #2:  Let the sides be 5, 6 and 5.  The perimeter is 16, so s is 8.  The area is as follows.

sqrt(8(8-5)(8-6)(8-5)) =
sqrt(8 × 3 × 2 × 3) =
12 (square units)

This takes more work that A = ½bh, but we aren't always given the height of a triangle.

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Here is a completely different way to define a triangle, three points on the xy-plane where one of the points is the origin (0, 0) and the other two are (x1, y1) and (x2, y2).  Now the formula for area is ½|x1y2 - x2y1|.  Let's do some examples of this.

Example #1: (0, 0) (3, -1) and (4, 2)
The area is ½|3×2 - -1×4| = ½|6 - -4| = ½|10| = 5 (square units) 


Example #2: (0, 0) (6, 3) and (4, 1)
The area is ½|6×1 4×3| = ½|6 - 12| = ½|-6| = 3 (square units) 

Practice for slope, distance and area of a triangle defined by three points.

We will deal with the following three points (3, 4), (7, 5) and (-2, 8) as well as the origin (0, 0)

Slope
a) slope from (3, 4) to (7, 5)
b) slope from (3, 4) to (-2, 8)
c) slope from (3, 4) to (0, 0)


d) slope from (7, 5) to (-2, 8)
e) slope from (7, 5) to (0, 0)
f) slope from (-2, 8) to (0, 0)

Distance




g) distance from (3, 4) to (7, 5)

h) distance from (3, 4) to (-2, 8)
i) distance from (3, 4) to (0, 0)


j) distance from (7, 5) to (-2, 8)
k) distance from (7, 5) to (0, 0)
l) distance from (-2, 8) to (0, 0)

Area of triangles
m) area of triangle with points (3, 4), (7, 5) and (0, 0)

n) area of triangle with points (3, 4), (-2, 8) and (0, 0)
o) area of triangle with points (-2, 8), (7, 5) and (0, 0)
p) area of triangle with points (3, 4), (7, 5) and (-2, 8)

 Answers in the comments.
 
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Simplifying square roots and square roots in fractions.

The editing software for the blog does not make it easy to write a square root sign (also known as a radical), so instead I will write sqrt(n) to denote the square root of n.

Square roots of perfect squares

Since 5x5 = 25, which we can also write as 5² = 25, we can simplify sqrt(25) = 5. Any perfect square inside a square root is written more simply as the original number before it was squared then the root was taken, assuming the original number was not negative.

Simplifying the square roots of numbers that are not perfect squares
Consider sqrt(24), which is not a perfect square.  We can find the prime factorization of 24 by a factor tree, a method shown in class, and get that 24 = 2×2×2×3.  If we take sqrt(2×2×2×3), we have a square, 2×2, under the root sign, so the pair may be removed and one is put outside the root sign.  (Here I used the metaphor taught to me so long ago by mean Mrs. Kruger, that the root sign is like a prison, and to escape you need a twin.  One twin escapes and the other is killed, never to be seen again.)  Using the prison break rule.  sqrt(2×2×2×3) = 2×sqrt(2×3) = 2sqrt(6).

Simplifying fractions with square roots
To put a fraction involving a square root in lowest terms, the denominator must be free of radical signs.  For instance, 1/sqrt(n) is not in correct form, so we multiply top and bottom by sqrt(n) and get sqrt(n)/n.

Example: 1/sqrt(8) can be changed to sqrt(8)/8, but 8 = 2×2×2, so sqrt(8) is 2sqrt(2).  Since the 2 in front of the radical is not "in the prison", we can reduce the fraction 2sqrt(2)/8 = sqrt(2)/4.

Practice problems

a) sqrt(50)
b) sqrt(48)
c) sqrt(99)
d) sqrt(98)
e) 1/sqrt(48)
f) sqrt(98)/sqrt(50)

Answers in the comments.

Practice with side lengths and classifications

Here is a list of triples, three numbers in a set.  Determine if the three numbers could be the lengths of triangle sides, and if so, classify the triangle by both classification types,

Classification by biggest angle: Obtuse, Right, Acute
Classification by side length relations: Isosceles, Equilateral, Scalene

a) 6, 3, 3

b) 7, 4, 1

c) 5, 5, 2

d) 5, 4, 3

e) 4, 4, 4

Answers in the comments.

Practice with two angles of a triangle.

In the following problems, two angles of a triangle are given.  Find the missing angle and find the classifications of the triangle based on angle relations (isosceles, equilateral, scalene) and on largest angle (obtuse, right, acute).

1) 70°, 10° ____________ Classifications: ______________ and _____________

2) 70°, 20° ____________ Classifications: ______________ and _____________

3) 70°, 30° ____________ Classifications: ______________ and _____________

4) 70°, 40° ____________ Classifications: ______________ and _____________

5) 70°, 50° ____________ Classifications: ______________ and _____________

Answers in the comments.

Fact about triangles

A triangle is defined by three points, sometimes called vertices (plural of vertex) connected by three line segments, usually called sides.  Two sides meeting at a vertex create an angle, so a triangle also has three interior angles.

The interior angles of a triangle always have a sum of 180°.  This means that if two angles are given, the third can be found using subtraction.

Classification of Triangles.

One classification method for triangles is determined by the largest of the three angles.

If the largest angle is greater than 90°, the triangle is obtuse.


If the largest angle equals 90°, the triangle is right.


If the largest angle is less than 90°, the triangle is acute.

A second method of classification deals with how the angles relate to one another.

If at least two angles have the same measure, the triangle is isosceles.


If all angles have the same measure, the triangle is equilateral. (In this case, all angles are 60°.)

If all angles are different, the triangle is scalene.

An equilateral triangle is a special case of isosceles.

If a triangle is defined by side lengths instead of angles, it is easy to tell which classification is true.

All sides the same length means equilateral.

At least two lengths are the same means isosceles.

All different lengths means scalene.

The Triangle Inequality

If we choose three positive numbers at random, it might be that these cannot be the side lengths of a triangle.  Since these lengths are straight line segments and therefore the shortest distance between two points, the other two lengths have to add up to at least as much as the long length.  (Think of the three points.  If I travel from A to B, it cannot be a short cut if I instead travel from A to C to B.) Likewise, the difference between two distances has to be less than third distance.

If we let the three sides be labeled a, b  and c,  we can state the Triangle Inequality as follows.

a + b > c > |a - b|

If I pick three points at random on a plane, they might all be on the same line.  So if b is directly between a and c, we need to take that in account by changing the greater than signs into greater than or equal signs.
 
a + b >= c >= |a - b|

Some notes about symbols used in trigonometry

Trigonometry is very closely connected to the geometry of triangles, so we will use illustrations like the one above on a regular basis.  Because there are so many different things to consider, we use different alphabets to signify different types of objects.

The standard way to discuss this particular triangle is to call it triangle ABC, where the capital letters in the English alphabet refer to the points.  (A point may also be called a vertex.  The plural is vertices.)  The sides are designated by lowercase English alphabet letters. The side opposite a lettered vertex will share the same letter, only in lowercase.  In this example, side a is opposite point A, side b is opposite point B and side c is opposite point C.

Angles are yet another different object and they are designated with Greek letters.  This blog's editing software is based on HTML, which doesn't have the Greek alphabet symbols readily available, so I will have to type out the words alpha, beta and gamma when referring to angles here.  On any other printed material, the actual Greek letters will be used.  As you can see in this picture, the angle at point A is alpha, the angle at point B is beta and the angle at point C is gamma.

We don't use all the Greek letters angle designations.  One obvious example is pi, which is the Greek letter for p in English and is already being used to mean 3.14159..., the ratio of the circumference of a circle to its diameter.  Besides alpha, beta and gamma, the most common Greek letters used in trigonometry are delta, which looks like a squiggly lowercase d, and  theta,  which looks like the letter o with a horizontal line crossing in the middle.

Syllabus for Math 50 for Fall 2011

Math 50: Trigonometry Fall 2011 – Laney College
Instructor: Matthew Hubbard
MWF: 9:00-9:50 am
Email address: mhubbard@peralta.edu
Recommended Texts: Trigonometry (open source) http://mecmath.net/trig/trigbook.pdf
Barnett et al, "Analytic Trigonometry with Applications" (10th Edition)
Office hours: MWF: 8:30-8:55 am G-210 (Classroom)

Add and drop class dates
Last date to add: Sat., Sept. 3
Last date to drop class without a “W”: Sat., Sept 17
Last date to drop class with a “W”: Wed., Nov. 23

Holiday schedule for MWF schedule
Labor Day: Monday, Sept. 5
Veteran’s Day: Friday, Nov. 11
Day after Thanksgiving Friday, Nov. 25

Test dates:
Midterm 1: Friday, Sept. 23
Midterm 2: Friday, Oct. 28
Comprensive Final: Wednesday, Dec 14 8:00-10:00 am

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%
Midterm I 25%*
Midterm II 25%*
Quizzes 25%*
Final 25%

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 three homework assignments.
The extra 5%: Extra credit will be factored into quizzes, midterms and homework at about 5%. This means 100 point tests with have 105 points and homework and quizzes will average 21 points each, though they will count as 20 points.
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.

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.

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.