MAT1275 College Algebra and Trigonometry, Spring 2017

Topics:

• Please make sure that you review the material on Trigonometry from the last two classes. You must become very familiar with the two special right triangles, and with the standard position for a triangle in the coordinate plane.

• Reflecting a right triangle in standard position: (this was done last time, but I’m mentioning it here because we make use of it today)

We can take a right triangle in standard position and reflect it over the x-axis or over the y-axis or both, to get three other triangles in the other three quadrants. These triangles are not in standard position, but they do have one vertex at the origin, and their right angles are on the x-axis (although not necessarily on the positive x-axis). We are going to make use of these reflected triangles in studying trig on the coordinate plane: they will be called reference triangles.

You can see some reflected right triangles in this Math is Fun post on the coordinate plane.

We reflected the 5-12-13 right triangle into the four quadrants and found the coordinates of the vertex which is not on an axis in each case. The coordinates of all four points are the same except for their signs, which depend on the quadrant.

• Angles in standard position:

An angle in the coordinate plane is said to be in standard position if its vertex is at the origin, and its initial side is on the positive x-axis. (Remember that we think of angles as rotations!) The angle is positive if the rotation is counter-clockwise, and the angle is negative if the rotation is clockwise.

• Coterminal angles:

Angles in standard position are called coterminal if they have the same terminal side. Coterminal angles differ by adding or subtracting a full rotation (360 degrees, or ) one or more times. From the definitions of the trig functions in the coordinate plane, it is clear that coterminal angles will have the same values for the trig functions. This shows that the trig functions are periodic.

• Radian measure of angles:

See the Math is Fun webpage on this topic and more discussion of radians, and also their exercises which are at the bottom of that second link! Very useful!

• The coordinate plane definitions of the trig functions: 

These are given in the outline notes that I handed out in class. This handout is also available here:

MAT1275TrigFns3definitions

Also see the Math is Fun post on the coordinate plane.

• The signs of the trig functions in the four quadrants: (the Math is Fun post also discusses these)

These are easy to see from the coordinate plane definitions, since the sign of the sine will be the same as the sign of the second coordinate b, and the sign of the cosine will be the same as the sign of the first coordinate a. (Notice that I am calling the two coordinates a and b, because we will be using x to stand for the angle later on!) We can then get the sign of tangent by using the identity .

I do not advocate using mnemonic devices to remember the signs, but rather, think about the signs of the coordinates in the four quadrants and how the coordinates are connected to the trig functions. This is a much stronger way to get all of those concepts connected in your memory!

The importance of knowing the signs of the trig functions in the four quadrants is that the trig functions of any angle are the same as the trig functions of its reference angle (which we will discuss next time), except for the sign.

Also, if we know one trig function of an angle and we know which quadrant the angle is in, we can compute the values of all the trig functions for that angle. We did two examples of this in class: here is one.

Example: If and $\cos(\theta) < 0$, what are the values of the other five trig functions?

• The unit circle definitions of the trig functions:

This is just a version of the coordinate plane definitions, where we take the radius  r to be 1. These definitions are also given in the handout.

• Very important identities which are easy to prove using the unit circle definitions:

(The Pythagorean Identity)

There are two more versions of the Pythagorean Identity that are easy to derive from this one: we will discuss them later.

Finding values of the trig functions for angles related to the special right triangles, using the unit circle definitions and reflecting the special right triangles.

You should practice until you are able to come up with the values of the trig function for these angles very quickly.

Homework:

• Review the material, definitions and vocabulary, and the examples discussed in class. Make sure that you know all the vocabulary, and the three definitions of the trig functions. Follow the links in the notes above to help you review!

• Do the WeBWorK: not due until Sunday evening because of Test 3, but don’t delay! There are 3 parts (so far).

• Also do the following problems:

1. Draw a right triangle whose legs are 3 and 4 in standard position in the coordinate plane, so the side of length 3 is along the x-axis. Then reflect that triangle over the x-axis and the y-axis and both axes to get three other triangles in the three other quadrants. Find the coordinates of the vertices which are not on the axes for those four triangles.
2. Find the values of the six trig functions if and . Simplify them completely.

Test 3 review self-tests have been posted on Piazza, along with the answers and references to the textbook sections. For completing the square in the equation for a parabola, you should look at the WeBWorK assignment on parabolas (which uses a different method than the textbook does, and is what we did in class.

Also posting these here:

MAT1275Test3ReviewSpring2017

MAT1275Test3ReviewAnswersSpring2017

Topics:

• Trigonometry topic: special right triangles (continued)

The two important right triangles are:
• The isosceles right triangle,
• The half of an equilateral triangle – which I will call the half-equilateral triangle for short.
These are commonly called by other names which refer specifically to the degree measure of their angles, but it is much better to call them by these names for two reasons (at least):
• Later we will use the radian measure of the angles, so we do not want to tie ourselves too much to degree measure
• The names given above remind us of what these triangles really are, so that we can recreate them if we forget the relationships of their sides, for example.

Important versions of the two special triangles: when the hypotenuse is 1.

You can see these two right triangles on this blog post from squarerootofnegativeoneteachmath (which also shows how they are used in the unit circle definitions of the trig functions, which we will discuss next time).

Embedding a right triangle in the coordinate plane:

We pick out one of the acute angles in the right triangle: let’s call it angle . We say that the right triangle is in standard position in the coordinate plane if the vertex of angle is at the origin, the side adjacent to angle is on the positive x-axis, and the hypotenuse extends into the first quadrant. This picture shows a right triangle in standard position. Notice that the coordinates of the top vertex are (x, y) where x and y are the lengths of the two legs of the right triangle.

We are going to use this embedding to extend the definitions of the trig functions to angles which are not acute angles, so they are not angles of right triangles.

Homework:

• Review the isosceles right triangles and how we found the length of the hypotenuse using the Pythagorean Theorem, and also the half-equilateral right triangle and how we found its height using the Pythagorean Theorem. If you do this a few times, you will end up memorizing the triangle! You must learn it by heart as we will use it (and the other special right triangle) a lot in Trigonometry. Here is a good web source that shows how these triangles were developed. (Do not get too attached to the degree-measure names, though!)

Here is another good source on the half-equilateral triangle. (This one was also linked above in the notes.)

• Please note that we will be working from the Trigonometry textbook for most of the remainder of the semester.

Remember that you can use the Piazza discussion board to ask questions if you get stuck on any of the WeBWorK or the other homework problems. Don’t forget to include the problem itself in your question, as that will make it easier for you to get a quick response!

Topics:

• Solving systems of linear equations: what does “linear” mean? Classification according to the number of solutions.

The pictures I used in class came from these two sites:
Systems of linear equations at Math is Fun

Solving linear systems in three variables

The basic classification is this:

Inconsistent system: has no solutions

Consistent, independent system: has exactly one solution

Consistent, dependent system: has infinitely many solutions

• Solving nonlinear systems using substitution and using elimination

here is the handout outline notes with the two examples we worked on in class:

MAT1275NonlinearSystemsNew

You can use them to rework those examples if you want.

Pay attention to choosing the method you will use when solving nonlinear systems. Also remember to be very careful to separate out the multiple solutions as you go!

• Trigonometry topic: special right triangles

We only discussed the isosceles right triangle (and this is the name you should use, rather than any name that refers to the degree measures of its angles, since we will be using another measure (radians) rather than degrees!

The two important right triangles are:
• The isosceles right triangle,
• The half of an equilateral triangle  (to be discussed next time) – which I will call the half-equilateral triangle for short.
These are commonly called by other names which refer specifically to the degree measure of their angles, but it is much better to call them by these names for two reasons (at least):
• Later we will use the radian measure of the angles, so we do not want to tie ourselves too much to degree measure
• The names given above remind us of what these triangles really are, so that we can recreate them if we forget the relationships of their sides, for example.

Homework:

• Review solving nonlinear systems. It is a really good idea to graph your system on Desmos (when you have access to it) so you can check that you have the right number of solutions, since that can be rather challenging to predict otherwise!

• Do the WeBWorK on this topic – it is not due until Sunday evening, but don’t put it off! Do at least some of it now.

• Review the isosceles right triangles and how we found the length of the hypotenuse using the Pythagorean Theorem. If you do this a few times, you will end up memorizing the triangle! You must learn it by heart as we will use it (and the other special right triangle) a lot in Trigonometry.

• Start working on the WeBWorK on that topic also. (You may not be able to do it all yet.)

• From the Algebra textbook, do the assigned problems from Section 9.4

• Read this web page on the isosceles right triangle and work the Review problems at the end of it.

• Please note that we will be working from the Trigonometry textbook for most of the remainder of the semester.

Remember that you can use the Piazza discussion board to ask questions if you get stuck on any of the WeBWorK or the other homework problems. Don’t forget to include the problem itself in your question, as that will make it easier for you to get a quick response!

Topics:

• Completing the square to put the equation of a circle into its standard form

• Brief review of solving a system of 2 linear equations in 2 variables by elimination

• Solving a system of 3 linear equations in 3 variables

The two outline notes pages which I handed out in class are also available here:

MAT1275systemsReviewClasswork

MAT12753by3systemsClasswork

There is a very nice video explaining this method at Patrick’s Just Math Tutorials. He explains the reasoning along the way.

There is a longer video explanation of the method along with background information about 3 by 3 systems at Khan Academy.

Homework:

• Review completing the square for the equation of a circle. Make sure also that you know how to read off the center and vertex of the circle, how to find four points on the circle from that information, and how to use them to draw the graph of the circle.

Free graph paper that you can print out is available from many sources online, for example at Print Free Graph Paper. This is very helpful in making sure that your graphs really do look like circles!

• Review solving 2 by 2 systems of linear equations by elimination. You should know how this method works, and also be aware which variable it is that you are eliminating. We need to know that when we apply elimination to solving a system of 3 equations in 3 variables!

• Finish working the handout (the 3 by 3 system) if you did not have time to do this in class. Make sure to check your answer by substituting back into all three equations! If you get stuck on this, please post a question to Piazza, and in your question post a clear photo of the work you did.

• Do the WeBWorK: start as soon as possible to do a bit of it, and do distributed practice!

• From the textbook, do the following problems: pp. 753-754 #39-47 odd; p. 286, #11, 13, 15. When you solve the 3 by 3 systems of equations, don’t forget to check your solutions by substituting back in all three equations.

Remember that you can use the Piazza discussion board to ask questions if you get stuck on any of the WeBWorK or the other homework problems. Don’t forget to include the problem itself in your question, as that will make it easier for you to get a quick response!

(Wednesday after the test)

Topics:

• Completing the square to put the equation of a parabola into its vertex form , when the leading coefficient is not 1.

Reminder: The method we use, which the WeBWorK leads you through step-by-step, is not the method used in the textbook.

Here is a video from VirtualNerd which shows the method being used when the leading coefficient is 1. I have not been able to find a source that shows our method when the leading coefficient is not 1: the best thing to do is to work through the WeBWorK exercises (you can do this even if the assignment is closed) and make notes on the steps.

• Drawing the graphs of quadratic functions (parabolas).

To have a complete graph, you should show the following features;

• The vertex

• The axis of symmetry – you do not have to draw it in as a line, but you should make it clear that your graph is symmetric around a vertical line through the vertex

• The y-intercept: this is the point where x=0.

• The x-intercepts, if there are any. (They are the points where y=0.)

I also showed you a little trick that you can use to get two other points of the graph starting with the vertex, and using the leading coefficient a. My notes are here: MAT1275graphingParabolasPictures

Remember the effect of the number a on the graph: the parabola will be stretched away from the x-axis or compressed toward the x-axis depending on whether |a| is greater than 1 or less than 1 (between 0 and 1), and the parabola will open downward if a is negative.

• New topic: The Distance Formula

The distance formula is a result of the Pythagorean Theorem, and it will be easier to remember it correctly if you understand how we get it from the Pythagorean Theorem. Here is a nice video explanation from Khan Academy.

We use the distance formula to get to our next topic, which is

• Circles

Because the definition of a circle is that it is the set of all points in the plane which are at a fixed distance r away from its center (h, k), we can get an equation for a circle using the distance formula:

Or by squaring both sides,

This is the standard form of the equation of a circle with radius r and center (h, k).

In case the center is at the origin (0,0), the equation will have a particularly simple form:

We looked at Examples 1 and 2 in Chapter 9, where we had to read off the center and radius of the circle and use them to draw the graph. To help draw the graph we find four points on the circle: starting at the center, go up by the amount r, then down by the amount r, then right by the amount r, then left by the amount r. This makes it easier to draw a nice circle.

Homework:

• Review the examples we discussed in class. Make sure that you understand the derivation of the distance formula and the formula for the circle, as well.

• In Chapter 7, do problems #45-51 odd

• In Chapter 9, do the assigned problems from the Course Outline up to #37 only

• Do the WeBWorK: Two very short assignments due by Tuesday 11 PM, but do not wait to the last minute!

Remember that you can use the Piazza discussion board to ask questions if you get stuck on any of the WeBWorK or the other homework problems. Don’t forget to include the problem itself in your question, as that will make it easier for you to get a quick response!

Topics:

• The Quadratic Formula and what we can learn from it about solutions to quadratic equations

• Graphs of quadratic functions

Summary of what the Quadratic Formula tells us about the solutions to quadratic equations: These observations come from the basic observation that the quantity under the radical sign (the radicand) is the part of the quadratic formula that can give rise to solutions which are irrational or complex. That is why we give that radicand a special name:

is called the discriminant of the quadratic equation .

What the discriminant tells us about the roots (solutions) : if all of the coefficients are integers, then

• If , there will be two solutions, both of which are real numbers.

• In the case where  and it is a perfect square, then the solutions will be rational numbers. [This means also that you could have solved the equation by factoring and using the Zero Product Property.]

• In the case where  and it is not a perfect square, then the solutions will be irrational numbers.

• If , there will be one solution, a real number.

• If , there will be two non-real complex solutions, which are conjugate to each other.

Graphs of quadratic functions:

We first considered the basic graph of . This graph is a parabola with vertex at (0,0), and it is symmetric by reflection over the y-axis. Please note that the graph also contains the points (1,1) and (-1, 1). You can see the graph of here (in desmos)

This is an example of a function : the idea is that we think of inputting a number for x, and the formula outputs a number ($latex x^{2}) which is the value of the function for that x. So x is the input and y is the output. (These are sometimes called the “independent variable” and the “dependent variable”.)

We then considered what would happen if we change the function by adding a number c to its output. So we are considering the graphs of functions of the form . By using desmos to play around with different values for c, we found that this moves the graph upward or downward by the amount c, depending on whether c is positive or negative. The graph with sliders to change the value of c is here (in desmos).

We next considered what would happen if we change the function by adding a number c to its input, in other words, we substitute (x+c) in place of x in the formula for the function. So we are considering the graphs of functions of the form . By using desmos to play around with different values for c, we found that this moves the graph to the left or to the right by the amount c, depending on whether c is positive or negative: it moves to the left if c is positive, and to the right if c is negative. The graph with sliders to change the value of c is here (in desmos).

Finally, we considered what would happen if we change the function by multiplying the output by a number a, so we are considering the graphs of functions of the form . By using desmos to play around with different values for a which were positive, we found that this stretches or compresses the graph toward the x-axis, depending on whether a is greater than 1 or a is between 0 and 1 . If a is negative, the effect is the same (stretching or compressing), but also the graph is reflected (“flipped”) over the x-axis, so it opens downward.  The graph with sliders to change the value of a is here (in desmos).

Putting these all together, we come up with a quadratic function of the form (using different letters now)

Compared to the basic parabola , this will be moved up or down by k units, right or left by h units, and it will have vertical stretching or compression due to the a. If a is positive, the parabola will open upward; if a is negative, the parabola will open downward. It vertex will be at the point (h, k). Here is the graph in desmos with sliders so you can change the values of a, h, and k and see what happens.

[Note that we took care of the problem that a positive number added to the input was moving us to the left, by writing (x-h), so instead of adding a positive number we are subtracting a negative number h in that case.]

Our object will be to take a quadratic function which is in the form and use completing the square to rewrite that equation in the standard form . Then we will easily be able to read the vertex of the parabola from the equation and also we will be able to sketch the graph very quickly and accurately by hand.

Please note that the method we use to rewrite the equation is NOT the method used in the textbook! In order to make it a bit easier to complete the square, we are first isolating the terms which have x in them on one side of the equation, and only at the end we go and solve for y again.

The WeBWorK assignments are related to the two things above.

Homework:

• Make sure that you have done all of the homework problems assigned last time on using the Quadratic Formula. (p. 595 #5-25 odd). Now go back to each of those problems, compute just the discriminant , and see how the number and nature of the solutions correspond to the discriminant as described in the summary above.

• Do the WeBWorK before you do the problems in the textbook below! There are two parts to the assignment, and the one on vertices will walk you step-by-step through the procedure for putting the equation into standard form . We did some of this in class.

• In the textbook, do the following problems: p. 625 #17-27 odd. Make sure that you are using completion of the square in order to find the vertex, not any other method. See Examples 1, 2 in Section 7.5, or see here. (link to PurpleMath – she completes the square using the same method we do, and explains how to read off the vertex and what the effect of a is on the graph.)

Remember that you can use the Piazza discussion board to ask questions if you get stuck on any of the WeBWorK or the other homework problems. Please be as specific as possible about what you are stuck on and don’t forget to put the problem itself into your question, as we may not have the textbook with us when we go to answer it!

Here is the Test 2 review Self-Test (it is broken into two parts, which you should use separately to test yourself):

MAT1275Test2ReviewSpring2017

And here are the answers along with references to review:

MAT1275Test2ReviewAnswersSpring2017

Topics:

• More on using the Square Root Property (from discussion of homework problems)

• Perfect squares of binomials (perfect square trinomials) and Completing the Square

• Quick look at solving quadratic equations by completing the square, to get to…

• Derivation of the Quadratic Formula

• Using the Quadratic Formula to solve a quadratic equation (more next time on this)

I am working on notes on the derivation of the Quadratic Formula and will post them here as soon as they are done.

Completing the square is not a method that is commonly used to solve a quadratic equation (and we will not really use it for this purpose): its main role here is to allow us to show where the Quadratic Formula comes from. (The Quadratic Formula is actually a theorem of mathematics, and the derivation of the formula is the proof of the theorem!)

There is a questionable moment in the derivation, when we take the square root of , and the textbook (and most sources) just says that without further comment. But that is not necessarily true! It is not necessarily true, because could possibly be a negative number: we know that . It turns out that you get the same end result even in the case where is negative, but we should not pass over that without comment. My notes will show how it works out.

It is always important to remember that only if ! Otherwise you can get into a lot of trouble. We made this point when discussing the Square Root Property. There are a lot of sources that write it incorrectly, and even people who should know better do that. (I’m not going to write the incorrect version here, because I don’t want to put it into your heads.) Just remember that

For any real number NUMBER

If  ,

then    or  

Or, in shorthand,

Comment on using the Quadratic Formula: The Quadratic Formula can be used to solve any quadratic equation at all: in that respect it is superior to the other two methods (factoring and using the Zero Product Principle, or using the Square Root Property) which can only be used when the equation we are trying to solve has a certain form. However, it is always better to use factoring or the square root property if possible! So you should examine your equation to see if one of those two methods can be used before trying the Quadratic Formula (unless you are specifically instructed to use the Quadratic Formula). There are several reasons to prefer those other methods when they can be used: one reason is that human beings, as opposed to computing machines, tend to make errors when using the QF, either by not correctly identifying the numbers a, b, and c, or by not doing the operations in the correct order, or by not having the formula correct in the first place. Another reason is that it is important to practice factoring polynomials, because factoring is used for so many other things. The more practice you do, the more easily you will be able to factor when you need to do so. Another reason is that factoring or using the square root property, when it is possible to do so, will almost always be a lot faster than using the QF. The QF is very useful and important, but never forget that you have other options available!

Homework:

• Review the discussion of the homework problems. Make sure that you understand the corrections we made.

• Review completing the square. It is very important that you see how this is coming from the special product patterns

and

(which you should always be using when the occasion arises). It is much easier to remember how to complete the square if you know why you are doing what you are doing. And we will need to be able to complete squares for other purposes later in the course.

• Do the following from the textbook: p. 581 #21-32 all; p. 595 #5-25 odd

Whenever you use the Quadratic Formula, it is an excellent idea to actually write it out and say it out loud while you do so, before substituting numbers in. This will help fix it in your memory.

• Do the WeBWorK, which has two (short) parts.

Note: in the assignment on the Quadratic Formula, the problems are phrased in the following form: “Find the roots of the parabola (for example). Finding the roots (more correctly, the x-intercepts) means exactly the same thing as asking you to solve the equation .

Remember that you can use the Piazza discussion board to ask questions if you get stuck on any of the WeBWorK or the other homework problems. Please be as specific as possible about what you are stuck on and don’t forget to put the problem itself into your question, as we may not have the textbook with us when we go to answer it!

Topics:
• Solving quadratic (and other polynomial) equations using the Zero Product Principle (section 4.8, up to Example 4 – here is a source which discusses factoring by grouping, also known as the ac method, for the case where the leading coefficient is not 1; and here is a video from Khan Academy showing it being done. Please make sure that you do this by the correct method I showed in class and which is shown in these sources, and in your textbook – DO NOT use the incorrect variation which some one may try to show you!)
• Solving certain types of quadratic equations using the Square Root Property (section 7.1)
The Zero Product Principle (or Property, or Rule,…):
If A* B = 0,
then either A=0 or B=0

In order to use the ZPP, one side of the equation must be 0 and the other side must be in factored form. See Examples 1-2 in section 4.8.

The Square Root Property (or Principle, Rule, …): for any real number k,

If  ,
then   or .

In order to use the SRP, one side of the equation must be a real number and the other side must be written as a perfect square. See Examples 1-3 in Section 7.1. Next time we will extend this a bit.

Homework:
• Review the two important rules above, and the examples we discussed in class (and others mentioned above). Make sure that you see how the ZPP and SRP are being used!

• Do the assigned problems from sections 4.8 (all of them) and 7.1 (up to #9 only) in the  Course Outline

• Do the WeBWorK: due by 11 PM Tuesday, start early!

Remember that you can use the Piazza discussion board to ask questions if you get stuck on any of the WeBWorK or the other homework problems. Please be as specific as possible about what you are stuck on and don’t forget to put the problem itself into your question, as we may not have the textbook with us when we go to answer it!