Key Terms
- Determinant: a number computed from a square matrix that can help find the inverse
- Limit: the value a given function approaches in order to determine continuity
- Matrices: arrays of numbers arranged in rectangular or square form
Materials
- Copy of the unit circle
- Graph paper
- Internet access
- Paper
- Pencil
- Polar coordinate graph paper (optional)
- Protractor
- Ruler
Time / Length
- Two days to complete the work in this precalculus assignment
- Two weeks to deliver two quality presentations to a teacher and/or class
Assignment Instructions for Students
Step One
Let's get started! What are some actual applications of trigonometric graphs in the real world? Well, they can be used for studying sound acoustics, population growth, ocean waves, and automobile engines, just to name a few.
The six types of trigonometric graphs we will discuss are the:
These two graphs look sort of like ocean waves and are similar to one another, but only the sine graph goes through the point (0, 0).
Example # 1:
Let's graph a sine wave in radians.
Pick your points:
If x equals -2π or -6.28, then y equals 0
If x equals -3π/2 or -4.71, then y equals 1
If x equals -π or -3.14, then y equals 0
If x equals -π/2 or -1.57, then y equals -1
If x equals 0, then y equals 0
If x equals π/2 or 1.57, then y equals 1
If x equals π or 3.14, then y equals 0
If x equals 3π/2 or 4.71, then y equals -1
If x equals 2π or 6.28, then y equals 0
Now connect your points by drawing a smooth and curving continuous line. Well done!
These two graphs look sort of like a series of parabolas, with the ones above the x-axis opening upwards, and the ones below the x-axis opening downwards.
These two graphs look sort of like a series of curving lines, with the tangent lines appearing to move in a positive direction, one of which passes through the point (0, 0), while the cotangent lines appear to move in a negative direction.
Now go to the Desmos or another online graphing calculator and type in:
- y = sin x
- y = cos x
- y = sec x
- y = csc x
- y = tan x
- y = cot x
Now, do you clearly see the difference between the six functions? Well done!
Step Two
Part 1
By the way, remember your Cartesian coordinate system in which we have a point (x, y)?
Well, what if I told you there existed a similar polar coordinate system in which we have a point (r, θ)?
''Oh wow! Is this like one of those cool science fiction movies in which there exist two alternate, but similar worlds?''
Great analogy. Yes, in fact, you can even convert between the two methodologies.
Example # 2:
Convert point (5, 12) to polar coordinates.
First, find your r with this formula:
r = √(x2 + y2)
r = √ (52 + 122)
r = √ (25 + 144)
r = √ 169
r = 13
Next, find your theta with this formula:
θ = tan-1 (y/x)
θ = tan-1 (12/5)
θ = 67 degrees
Therefore, you have converted your point (5, 12) to the point (13, 67 degrees).
Now you go ahead and try one.
Problem # 1:
Convert point (6, 8) to polar coordinates.
Part II
Now let's demonstrate how to graph in polar coordinates. It's not too complicated. First, you just need to remember you will be graphing on a circular graph instead of on a rectangular graph. Second, you need to remember positive is counterclockwise (to the left) and negative is clockwise (to the right) on your graph.
Okay, do you have your graph paper, protractor, and ruler handy? Of course, this will be even easier if you have polar coordinate graph paper. If not, then simply draw a series of concentric circles on your regular graph paper.
Graph the following polar coordinate points:
Count five spaces to your right along the x-axis, and then use your protractor along with your ruler to create a line at a 45 degree angle from the x-axis. In Quadrant I, go ahead and plot your point A.
Count three spaces to your right along the x-axis, and then use your protractor along with your ruler to create a line at a 75 degree angle from the x-axis. However, since you have a negative sign, you will be drawing this line in Quadrant IV instead. Plot your point B.
This is a bit trickier. Since you have a negative sign, count four spaces to your left along the x-axis, and then use your protractor along with your ruler to create a line at a 30 degree angle from the x-axis. Please note this will be in Quadrant III, as you move counterclockwise for a positive number.
You get the idea. It's not much different from plotting rectangular coordinates, but you can get a bit confused on which way to move along your circles.
Step Three
Let's learn to determine the limit of a function.
For example, suppose we are asked to find the limit of the function:
Limit ((x - 3) ((x + 2)) / (x - 3)
As x → 3
Well, immediately we know our function is not continuous at x = 3, because when we plug 3 into the equation, we get 0/0 and we can't divide by zero (even when there is also a zero in the numerator).
Looking more closely at this, we see we can actually cancel part of our numerator with our denominator, which results in:
Limit x + 2
As x → 3
Now we simply plug in 3 to get our answer of 5.
It's your turn!
Problem # 2:
Evaluate:
Limit (√x - 3) / (x - 9)
As x → 9
Step Four
Lastly, the term matrix refers to an array of numbers that is used to help solve systems of equations sometimes involving real-world situations.
First, let's solve to obtain the determinant of a matrix, which is a scalar value that provides us with relevant information.
Let's say you are given the 2 x 2 square matrix:
Then your determinant will be:
a * d - b * c
Problem # 3:
Given the square matrix:
Find the determinant.
Deliverables
You can choose either Option A or Option B to demonstrate your command of this material.
Option A
If you choose this option, count it as a double deliverable, and then you don't need to design a second deliverable:
- Use all the topics discussed in this lesson (limits, matrices, polar coordinates, and trigonometric graphs) to create challenging questions in the form of either a Wheel of Fortune or Jeopardy game show which include detailed answer explanations. There are free online templates for these games or you can work with index cards and a bulletin board. Ideally, you will need other students to play your game and a SmartBoard to make this idea work well. It will take time and skill to put together quality math questions. You can also use office software or online resources to help you make the graphs and math equations.
Option B
If you go with this option, pick a topic we learned about (limits, matrices, polar coordinates, and trigonometric graphs) and one of the four project choices below.
Because these are shorter projects than Option A, you will need to complete a total of two of these possible projects, each using a different topic and choice of project.
- Design a synoptic poster or wall mural which contains information on your topic as it is covered in this assignment, including original practice problems.
- Explain one of the concepts to another person via your own YouTube video or TED Talk podcast. You may go into more detail, and you need to use your own examples.
- Make a comprehensive animation to clearly show one of these precalculus skills.
- Write summative questions to assess understanding of these mathematical concepts and processes. (For this option, create a summative question for each of the four topics).
Answers
Solution # 1:
Convert point (6, 8) to polar coordinates.
Here is the handy formula.
r = √(x2 + y2)
r = √(62 + 82)
r = √(36 + 64)
r = √100
r = 10
Next, let's find theta, and we have yet another fun formula:
θ = tan-1 (y/x)
θ = tan-1 (8/6)
θ = 53 degrees
You have converted your point (6, 8) to point (10, 53 degrees), and you're done!
Solution # 2:
Evaluate:
Limit (√x - 3) / (x - 9)
As x → 9
First of all, we see the indeterminate case will be at 0/0 since we know we can't divide by zero.
Next, let's get rid of the radical in the numerator by multiplying both our numerator and our denominator by our old friend the complex conjugate:
((√x - 3) * (√x + 3)) / ((x - 9) (√x + 3)) =
(x - 9) / ((x - 9) (√x + 3)) =
Cancel to obtain:
1 / (√x + 3) =
Finally, plug in your 9 for x to see the limit equals:
1/6
Solution # 3:
Given the square matrix:
Find the determinant.
Well, you just learned your determinant is:
a * d - b * c
So:
6 * 3 - 5 * 7 =
18 - 35 =
Therefore, your determinant is -17.
Assignment Rubric
| Requirements |
0 - 5 points |
| Problem # 1 is correctly solved |
|
| Problem # 2 is correctly solved |
|
| Problem # 3 is correctly solved |
|
| Subtotal |
|
| Deliverable(s) 0 - 25 points |
|
| Total: |
/ 40 points |
