Graphing Trig Functions Examples

If you  missed class today, here are a few examples on how to graph different types of trig functions, as shown in class. Please click on the video links below for a tutorial on how to graph. Happy studying!

How to graph a horizontal translation:   y = sec(x - pi/4)


How to graph a vertical translation: y = cosx + 2


How to graph a change in amplitude:  y = 3cscx


How to graph a change in Period:   y = tan2x

Review Week for the Semester Exam (12/11 through 12/17)

Hello trig students! Here is what we are reviewing for the trig exam next week:

Wed 12/11: Triangles Review (Pythagorean, SOHCAHTOA, LoS, LoC)
Thurs 12/12: Six Trig Functions Review (point, line, unit circle, quadrantal)
Fri 12/13: Reciprocals and Word Problems with Triangles
Mon 12/16: Graphing Trig Functions
Tues 12/17: Arcs/Sectors (6th Period Only, I will not have 2nd or 4th today)
Wed 12/18: Wrap-up, any last minute questions, turn in review sheets

Instead of posting review information again, scroll through this blog and find examples and notes of problems that you may have forgotten how to do. There are a few videos that might be able to help you out. Finally, if you need additional assistance, COME ASK ME during a time that we both are free! Good luck and happy studying!

Triangles Review: Which Triangle Rule to Use?

We have four different rules for triangles - how do we know which one to use? The information below may be helpful...

First, determine if your triangle is a right triangle or not. If you have a right triangle, you can use the Pythagorean Theorem or SOHCAHTOA. Use Pythagorean if you have two sides and are missing a third side. Use SOHCAHTOA if you have an angle measure and a side.

If you don't have a right triangle, you can use either the law of sines or law of cosines. If you have angle-side-angle, use the law of sines. If you have side-angle-side or side-side-side, use the law of cosines (for the angle, make sure you leave the variable next to cosine).

If you have side-side-angle, then the triangle cannot be solved, so don't use any triangle rules!

Law of Cosines

The law of cosines can be used in two circumstances. First, we can use it to find the missing side of a non-right triangle when we are given a side, an angle in the middle, and another side. Or, we can use it to find a missing angle of a triangle when we are given all three sides of a non-right triangle.

The law of cosines is:
c^2 = a^2 + b^2 - 2abcos(c) where a and b are the two shorter sides of the triangle, c is the longest side, and cos(c) is either the given angle or the missing angle (if you have all three sides).

It does not matter what side you label a or b, just make sure that c is your longest side!

Law of Sines in Real World Context

Today we practiced problems using the Law of Sines in real-life situations. In this case, we're not going to be given the triangle, we have to read the problem carefully and figure out where the correct angles and sides go. Also, remember from today, we discussed that when we make a triangle it is very important that:

<A is across from a
<B is across from b
<C is across from c

Otherwise, you're not going to set up the correct proportion (and thus, not get the right answer).

Law of Sines

We can use the Pythagorean theorem or SOHCAHTOA to solve for missing sides and angles of RIGHT triangles. What happens when we don't have a right triangle? We use the law of sines! The law of sines says that a/sinA = b/sinB = c/sinC. We typically only use one proportion at a time.

To use the law of sines, you must be given two angles and a side. Two sides and an angle doesn't work. First, find the third missing angle by subtracting your two given angles from 180, because there are 180 degrees in every triangle. Then, use the law of sines to create proportions to solve for your missing sides!

To see a video example of how to do this, click the link below:
http://www.educreations.com/lesson/view/law-of-sines/14387213/?s=9doa0j&ref=app

Trig Project BONUS Opportunity

For an extra 20 points added onto your final project score, you must go to a teacher in the morning, their service period, lunch, or after school, and present your project to them. It does not have to be on their smartboard - in fact, you can do it while sitting down with them on their computer. However, you must walk them through your presentation and explain to them how you came up with your graphs.

For the full 20 points, you must:

1) Present your full presentation to a teacher (the teacher can be me)
2) Get the teacher to EMAIL ME that you presented to them. no email = no bonus. This is SUPER IMPORTANT, make sure that this happens for you to receive your points.

Example of Trig Presentation

Hi Trig!

Please click this link here for the example trig project that I showed you in class:
Graphing_Trig_Powerpoint_Project.pptx - https://drive.google.com/file/d/0B3KJuak8r9sHR21hZlhZQzV6aDg/edit?usp=sharing

RULES: You cannot copy any of my examples!
Your work must be your own.
You must have all six trig functions
You must make your own modifications to the trig functions: one base function, one flipped function
one change in amplitude, one change in direction, one vertical translation, and one horizontal translation.

Deriving Equations from Graphs

So far, I have given you an equation and asked you to graph it. Today, we did the opposite. You saw a graphed equation on a coordinate plane and had to come up with an equation for it.

Basically, we want to identify the change in the graph and reflect that change in our equation. Is there a change in period? How much? And where does that number go in the equation? Is there a change in amplitude? If so, how much, and where would we find that in the equation? The same is true for vertical and horizontal translations.

This concept is rather difficult to explain in words so please click the video link below to get two good examples of how to do this.

http://www.educreations.com/lesson/view/deriving-equations-from-graps/13711498/?s=35CfqD&ref=app

Graphing cotangent

Graphing cotangent is JUST like graphing tangent - except backwards! Our graph goes up towards positive infinity near the -90 asymptote and down to negative infinity near the 90 asymptote. This is literally the ONLY difference between tangent and cotangent. Starting point, ending point, and period are all the same.

Since this is pretty easy to grasp, we did some harder examples of cotangent graphs in class today. Click on the link below to view a video of one if you are confused!

http://www.educreations.com/lesson/view/graphing-cotangent/13607137/?s=rkTVsd&ref=app

Graphing Tangent Functions

Tangent is different than sine, cosine, secant or cosecant because the graph does not appear in "wave" form. It also has a period of 180 degrees rather than 360 degrees, as the rest of our functions do. Its normal starting point is -90 and its normal end point is 90 degrees. Tangent repeats itself every 180 degrees.

To get a video tutorial on how to graph the base function of y = tanx as well as a vertical translation y = tanx + 2, click here:

http://www.educreations.com/lesson/view/graphing-tangent/13378530/?s=aEjnFr&ref=app

http://www.educreations.com/lesson/view/vertical-translation-tangent/13378707/?s=uuPIgT&ref=app

Graphing cosecant and secant functions

Cosecant and secant graphs are the opposite of sine and cosine graphs. Therefore, if we plot the sine or cosine graph as a reference, we can easily find the secant or cosecant graph as well!

Check out these links for videos on how to graph!
http://www.educreations.com/lesson/view/graphing-cosecant/13377826/?s=KKxlIw&ref=app

http://www.educreations.com/lesson/view/graphing-secant/13378142/?s=P3lOIB&ref=app


Note that the period, starting point, and ending points are ALL the same for sine, cosine, secant, and cosecant graphs. Amplitude is the only feature that changes (since the graphs are flipped).

Updates on the Status of the Graphing iPad Quiz

Hey Trig! I'm sorry for all of the confusion surrounding the iPad quiz. For 2nd and 4th periods: we will be ready to go for Monday so bring your 1/2 page cheat sheets and be ready to record. 6th Period: you were fantastic with those quizzes and I was genuinely impressed by the quality of work that you submitted. Well done! :)

Practice for Graphing Quiz, Day 1

We practiced graphing three trig functions on the iPads today. Your goal on the quiz is to be able to graph a function while explaining what you are doing while you work, and a brief explanation on why you're doing it. If you want additional practice, try going to the online website that we used today:

awwapp.com

which is an online whiteboard. Here, you can graph trig functions (or anything else, for that matter) and practice without killing the rainforest. Hooray!

Videos of me graphing and explaining trig functions

http://www.educreations.com/lesson/view/graphing-trig-function-example/12573134/?s=0Zzdf5&ref=app

http://www.educreations.com/lesson/view/graphing-horizontal-translations/12573232/?s=XYVdvD&ref=app

Two GREAT examples of how to graph trig functions with multiple numbers in them! Note: you must have the "Educreations" app downloaded to view these tutorials. to do this, just go to your App Store and search "Educations" and you can download it for free. You'll have to create a free account and then you can view all of my free tutorials!

Horizontal Translations of Trig Functions

This is our last type of change to a base trig function! We see horizontal translations when there are parentheses in the function, such as in f(x) = sin(x + 2/3pi). You will usually see the term inside the parentheses in radians, and since we don't like radians, you'd want to change it to degrees first then re-write the function using degrees: f(x) = sin(x + 120).

Horizontal translations change the start and end point of the function on the x-axis. We know that sine/cosine normally start at 0 and end at 360. So we form an inequality: 0 <= x + 120 <= 360 and solve to get x by itself in the middle by subtracting 120 from everything. This gives us -120 <= x <= 240. So, now the function begins at -120 and ends at 240! From there, find your new midpoints and graph your function.

Note that horizontal translations ONLY change the start and end points. They do not change the amplitude, period, direction, or starting point on the y-axis. There may be other numbers in the function that do that, but anything inside the parentheses is a horizontal translation only.

More Challenging Vertical Translations of Trig Functions

We see a vertical translation when there is a number behind the x-value in the function. However, there may be multiple numbers in the function that we need to address.

For example:
y = -2sin4x - 3.

All of the numbers in the function do different things to change sinx from its base function position.

-2, since it is in front of the sin, changes the amplitude of the function. Normally sine has an amplitude of 1, but this makes the amplitude -2. Instead of going up first, the negative flips it, and makes it go down first.

4 changes the period since it is in front of the x. We start with 2pi (always), and divide it by 4 to get the new period. Since we would get 1/2pi, convert this from radians to degrees by multiplying by (180/pi). This will give you a new period of 90 degrees.

-3 is the vertical translation since it is at the end of the function, behind the x. This means that instead of starting and ending at 0, as sin usually does, the function starts and ends at -3 on the y-axis.

Vertical Translations of Trig Functions

A vertical translation means that we move a base function (a function with no numbers) up or down on the y-axis. In other words, we change the starting and ending points for the y. The amplitude, period, and degrees on the x-axis stay the same - only the y-axis changes.

We see a vertical translation whenever there is a number added or subtracted at the end of the function. For example: y = sinx + 2 means that we would take the basic graph of y = sinx and move the whole function up 2 units on the y-axis. Instead of starting at (0, 0), we would start at (0, 2).

For cosine, it's a bit trickier because the graph of cosine begins at (0,1) instead of (0,0). Therefore, for the graph of y = cosx + 2, we would start at positive 3 instead of positive 2 because cosine already starts at 1, so we have to move the graph up two units from where it already starts.

IMPORTANT NOTE: just because we have a vertical translation, does not mean the amplitude of the function changes. Unless we see a number in front of the function, the amplitude would still be 1. However, if we do see a number in front of the function, such as y = 2sinx + 2, then we would have a vertical translation as well as a change in amplitude (because there is a number in front of the trig function).

Period of Sine and Cosine

We know that the base function of sine and cosine goes from 0 - 360 degrees. When we change the period of the function, we are changing how long/short the function is before it repeats itself again. So instead of going from 0 - 360, they would be going from 0 - some other number.

We know that a change in amplitude can be found be y = 2sinx where the 2 in front of the sin shows how high the new function goes. For a change in period, the number is instead directly in front of the x, such as y = sin2x. To get the new period, we know that 360 = 2pi. Divide 2pi by whatever is next to the x: in my example, it would be 2pi/2. The pi is left over, so pi would be our new period. We would much rather work in degrees than radians, so convert pi to degrees by multiplying by 180/pi. The pi's cancel, and you have 180 degrees left. Thus, 180 degrees is the new period of pi. The function still starts at zero, still ends at zero, and still has the same shape, but it becomes only half as long as before. The same rules apply for cosine functions, too!

Note that the amplitude doesn't change if there is no number in front of the trig function. Our function still goes from [-1, 1], but now, it just repeats itself more often because it's period is shorter.

Thursday and Friday: Amplitude for Sine and Cosine Functions

Now that we know what the base function for sine and cosine looks like, we can manipulate/change that base function in many ways. The first way is to change the range of the function by stretching/compressing it: this is called a change in amplitude.

When changing the amplitude of the function, we would expect to see a number in front of the trig function, like this:

y = 4sinx
y = (1/4)sinx
y = -4sinx

and the same applies for cosine functions as well. The "4" in the first example stretches the normal range of sine [-1, 1] to [-4, 4]. The "1/4" in the second example compresses the normal range of sine [-1, 1] to [-1/4, 1/4]. And the "-4" in the third example both stretches the function, like in the first example, and flips the function upside-down.

We practiced graphing on Thursday and made artwork out of five trig functions on Friday, where Mr. Williams will be judging for the top three on Monday. Shout out to everyone who brought their creative side to class on Friday! (and special shout out to Quadarius Gatewood, who found the creative side that he didn't think he had in a matter of 15 minutes!)

Base Functions for Sine and Cosine

Not only can we use sine and cosine to find missing sides of a right triangle, or points on the unit circle, but we can also graph them as functions of x and y on the coordinate plane. We graph them using degrees found on the unit circle. Sine's base function is y = sinx, and cosine's base function is y = cosx, and they look like this:

Both sine and cosine have infinite domains from negative infinity to positive infinity. Both of their ranges are from -1 to 1. And both repeat themselves after 360 degrees. When a function repeats, we call it a periodic function. Both of the periods of sine and cosine are from 0 - 360. The only difference between sine and cosine are that sine begins and ends at 0, while cosine begins and ends at 1.

Guest Speaker and Meeting out of Nowhere

We had a guest speaker yesterday from Arkansas State University! Shy the master's student told us about the Oral Communications class that she teaches to college freshman, and what it's like to be a freshman on the A-State campus. Expectations are a lot different in college - especially when your class only meets one day a week but you have it for 3-hours at a time!

I was pulled from my second period class today (just as we were about to graph y = sinx - boo!) to attend a training session at the Junior High. My apologies for this one - I didn't know about it in advance. A BIG THANK YOU to everyone who practiced their math ACT while I was gone and a HUGE Shout Out to my aid Bre Womack for setting everything up while the sub was there!

Error Analysis

All of my classes are close to being caught up - thank goodness! While 6th period was catching up on Friday, 2nd and 4th period did an exact trig values error analysis (shout out to 4th period for doing a great job explaining it when Mrs. Long walked in!). We looked at problems and found where the mistake was being made in each. Then you said how the problem should be fixed, and then fixed it to find the correct solution.

I have most of the review sheets. If I don't have yours yet, please get it to me so that you can at least get partial credit (remember, report cards come out soon, don't make a zero right before report cards!).

Negative Exact Values of Trig Functions on the Unit Circle

Finding a negative exact value of a trig function is very similar to solving when an angle is greater than 360 degrees.

We know that the unit circle begins at zero degrees and goes all the way up to 360 degrees. For anything outside of that range, we must use properties of coterminal angles to solve and find the corresponding degrees/coordinates on the unit circle.

Suppose that you were given an angle of -120 degrees. Since that's not on the Unit Circle, we would add 360 and finding its corresponding degrees and coordinates. We could then use those (x, y) coordinates to find the exact values of sine, cosine, tangent, cotangent, secant, and cosecant at that degree.

Review sheets due tomorrow for everyone except 6th period - yours are due Monday.

Cosecant, Secant, and Cotangent Exact Values, and Coterminal Exact Values

Yesterday, we discussed how to find exact values of sine, cosine, and tangent using values on the unit circle. Today, we did the same thing except with their inverses: cotangent, cosecant, and secant.

Since we know that, in a coordinate pair (x,y)
 sine = the y value
cosine = the x value
tangent = y/x

We can use the inverses and say that:
cosecant = 1/y
secant = 1/x
cotangent = x/y

Sometimes, the radian that we convert to won't be located within the 0-360 degrees that we see on a unit circle. For example, 13pi/3 converts to 780 degrees, so that would be way off of 360, which is the last degree measure on the unit circle. In this case, we must use properties of coterminal angles to solve. Simply subtract 360 until you get a measurement that is on the Unit Circle (in this case, 780's coterminal angle would be 60 degrees). Then use the (x, y) coordinates for the coterminal angle instead.

Finding Exact Values of Sine, Cosine, and Tangent on a Unit Circle

The unit circle is a circle on the coordinate plane whose radius is "1." We like the number 1 as a radius because it's super-simple and, generally, the easiest non-zero whole number to work with. The unit circle looks like this:

and we began class by making a reference picture to fill in all the exact coordinates of the unit circle. Will we ever be tested on any angle measures BESIDES the ones on the unit circle? NO. These are the most common and the only ones that you need.

From prior knowledge, we know that sin = y, cos = x, and tan = y/x. So to find the exact trig values for a given angle in degrees or radians, simply insert your y value if you're finding sine or your x value if you're finding cosine. Tangent is a little more tricky: you have to simplify y/x and divide your fractions. More often than not, something will cancel out.

Review for Quiz

2nd Period: We reviewed for the quiz and you took it in the same period - well done! Most of you did really well and I'm proud of your efforts. Shout out to those who stayed after the bell to make sure you finished your work.

4th and 6th Periods: Due to my poor planning and lack of time due to the pep rally, our quiz will be on Monday. We made the cheat sheets in class today - don't lose them! There are 5 quiz questions and 1 bonus question. You will need to know:

1. How to calculate an arc of a circle
2. How to calculate the area of a sector
3. How to work a problem with two interlocking arcs moving together
4. How to distinguish between arc and area questions.

HINT for the Review Sheet due next Friday 10/4

For Questions 5-7:

The number on the outside of the circle represents the length of the arc. Notice that you aren't given the central angle: you're going to have to solve to find it. Start with the formula for the length of an arc. Plug in the arc length and the radius, and solve for theta. Conveniently, this puts theta in radians, which is what we want it to be in anyway. Then, plus in the theta and the radius back into the formula for the area of the sector, and find the area (putting the correct units on at the end!).

You can use this strategy for all 3 questions.

Applications of Sectors

We did a weird experiment today with sector measurements, protractors, string compasses, and oatmeal creme pies (sorry if you missed it!). We cut away pieces of the oatmeal creme pie and measured the remaining part of the sector using the compass, ruler, protractor, and the formula for the area of a compass.

Don't forget that when the problem asks you for the REMAINING area of the sector, they are wanting you to do 360 - the given angle = the angle you want to use. This will be your central angle for the remaining sector. Then convert it into radians and plug it into the formula for the area.

Area of a Sector

While arcs can be thought of as a perimeter of a circle, sectors can be thought of as the area of (part of) a circle. It's formula is very similar to that of the arc formula:

where r represents the radius and theta represents the degree of the central angle in RADIANS not degrees (if it is in degrees, then convert it).  Remember that your final answer must include units and, since we're finding the area of something, the units must be squared.

Applications of Trigonometric Arcs - Airplane Travel

Our activity today was based upon using arc lengths to calculate distances across the Earth. Given that the radius of Earth is approximately 4000 miles, we can calculate the distance between two cities if we know the central angle (converted to radians, of course) between those two cities.

This turned into a group effort where students found distances to fly their planes while trying to get paid more than any other group. Fuel efficiency was taken into consideration and there was a travel limit based on the number of gallons that the plane began with.

Shout out to Jay Rice, Kyle Krus, and Chris Coleman for developing an algorithm to determine the MAXIMUM amount of money that could possibly be made given the amount of gas in the tank, the fuel efficiency, and the most profitable destinations - WELL DONE!

Another shout out to the group of QuaDarius Gatewood, Joe Tripp, and James Petty for earning over $100,000 with their plane.

Applications of Arc Lengths

We know that to find the length of an arc, we must multiply the radius of the circle times the interior angle of the circle (or central angle) so long as the central angle is in radians, not degrees. In circular trigonometry, we always measure in RADIANS. Make sure to convert first!

Today, we found the missing angle of a larger/smaller circle given the radius/central angle of another. To do this, we first found the missing arc length, then set it equal to the arc length of the other circle to solve for the missing central angle.

For a real-life application of this problem, we looked at gears on a clock and how they rotated together. Depending on the size of the gear, the gear rotated a different amount of total degrees (which we solved using the s = r times theta formula).

Review for the Angles and Triangles Test

Your first test is tomorrow! Remember that you are allowed to bring a 1/2 page cheat sheet of notes - front side only. There are 10 questions and 2 bonus questions on your test. Make sure that you show all of your work for each question.

The things you should be able to know how to do for this test are:

1. Solve trig equations
2. Simplify trig reciprocal identities
3. Find the six trig functions on a graph given a point or a slope
4. Find the missing sides of a right triangle
5. Find the missing angles of a right triangle (and convert them to radians for bonus)
6. Simplify expressions using properties of quadrantal angles.

Good luck and happy studying!

Solving Trig Equations

Solving trig equations are just like solving regular equations. First, do any distributing and combining like terms that you see. Get your thetas with your thetas and your numbers with your numbers - try and keep theta positive if possible. Then, simplify your equation until you only have one number left, and it's next to a trig function. Perform the trig operation, then use the inverse functions on the calculator to solve for theta, the missing angle.

Remember that you always want to have the SAME trig function in your equation when you solve. If you don't have this, you may have to use a reciprocal identity to simplify your equation first before you begin to solve.

Reciprocal Identities

An identity in math is something that is equal after both sides of the equation are simplified. Reciprocal identities simply mean the opposite of each trig function. We have three fundamental reciprocal identities in trig:

sin = 1/csc,    cos = 1/sec,    tan = 1/cot

We can simplify expressions using reciprocal identities by first flipping the numerator and denominator. Then, check to see if the radical is in the numerator, if not, then multiply by the radical to get it into the numerator (and out of the denominator). Next, check and see if the radical is in simplest terms. If not, break it down and take out any pairs of numbers that you find (anything that is not a pair is multiplied back together and stays under the radical). Finally, check to make sure that your fraction is in simplest terms. If not, reduce it. Don't forget to include the trig function as part of your answer!

Signs and Ranges

Our six trig functions have different signs depending on their location on the coordinate plane. In quadrant I, all trig functions are positive. In Quadrant II, only sine and cosecant are positive. In Quandrant III, only tangent and cotangent are positive. And in Quadrant IV, only cosine and secant are positive. This holds true for all triangles that we can draw on a coordinate plane, and will come into greater effect later on in the year.

Guest Speaker from Fayetteville/Mr. Clay

Dr. Watson, a petrochemical engineer from the University of Arkansas, stopped by the library today to talk about a two-day trip opportunity. However, if you are in 4th or 6th periods, then you mostly acquired your information through Mr. Clay, the career counselor. He also discussed ACT score and how important it is to the college application process. You need a 19 to get into the University of Arkansas unconditionally, and 18 to get in with remediation classes.

Remember, I do after school ACT prep from 3:40pm - 5:40pm in Mrs. Cain's room - 120. We cover all the subjects: Mrs. Cain does English and reading, I do math and science (aka how to read graphs properly). Come and join us - practice makes perfect!

Shout out to Chakara Adkins, Monica Horton, Chandler Baker, and Mark Robinson for coming to ACT prep today after school!

Thursday and Friday: Recreating the Mayan Sundial

Just as the Mayans used trigonometry to tell time with sundials, we can also use ourselves to determine what (relative) time of day it is. On Thursday, we went outside and measured our shadows while facing the sun. Students went home and got their own measurements at a different time of the day, using the same procedure as in class. Then, on Friday, we explored the relationship between the two triangles created at the two different times of day.

Our bodies represented the height of the triangle, our shadows were the base, and the hypotenuse was the distance between our head and the tip of the shadow. Students should have determined that shadows are longer at dawn/evening and the shortest right around noon. This is because the angle of elevation (that is, the angle between the Earth and the sun's position) is small during dawn/evening. Our shadow represents the adjacent side. The smaller the angle becomes, the larger cos(our angle) becomes, and thus, the longer the adjacent side (our shadow) becomes.

Shout out to Kierra Robinson and Joe Tripp for making over 100% on their trig projects - they were just THAT good!

Quadrantal Angles and Missing Angles

Quadrantal Angles end on the quadrants - they are angles that are multiples of 90 degrees. If one of the sides of the angle passes through a point on the axis, we cannot draw a triangle to calculate the six basic trig functions. Thus, we have to remember in this case that:

y = the opposite side
x = the adjacent side
the distance between the origin and the point: the hypotenuse

We also calculated missing angles within a triangle. This can be done by using the 2nd - inverse functions on the calculator above the three trig functions. We can fill in all missing sides and angles of a triangle.

The second review sheet was given out today, and you know how to do everything on it! Due date is September 12th, early due date is September 11th for 10 bonus points. Friday is the last day to retake the last quiz.

Trig Review for Quiz

Quiz tomorrow! Your quiz has 12 questions and 2 bonus questions. Study your notes and don't forget to bring your 1/2 sheet of notes as a reference sheet.

The examples that we did in class today will help you greatly with your quiz. If you can do those, you'll be just fine on the quiz.

Review sheets were collected today. If you didn't turn it in then I will take it tomorrow for half credit. I'm not taking any review sheets that are more than two days late.

The Six Trig Functions (2 days)

The Six Trig Functions are sine, cosine, tangent, cosecant, secant, and cotangent. We can use SOHCOATOA (sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, and tangent = opposite/adjacent) to help us solve for missing sides of right triangles. It is convenient if we already have a picture of the right triangle to help us, but sometimes, this is not the case. We may often need to draw the angle ourselves on a coordinate plane. We can do this using a point and the origin or using the origin along with an equation of a line.

Important things to remember when you are drawing your graph/angles:
-Begin in standard position, at zero degrees.
-Always rotate counterclockwise, in the positive direction, unless otherwise specified by the problem.
-Your angle (theta) always goes closest to the origin
-When drawing your triangle, draw it closest to the positive side in which you are rotating.
-Solve for a missing side using the Pythagorean Theorem. We don't necessarily have to know what the angle measure is in order to do this.
-Leave your answers in radical form. I prefer "the square root of 5" as opposed to 2.236. Why? Because radical notation is, mathematically, more accurate.

Bring your review sheets tomorrow! Quiz Friday!

Coterminal Angles in Radians

Thus far, we have measured everything in degree mode. The second way to measure an angle is in radians, which is in reference to the circumference of a circle. Radians always have the term π as part of the expression.

To convert from degrees or radians, we must multiply by either (π/180) or by (180/π). Use the first one to convert to radians and the second one to convert to degrees.

Finally, we made the trig unit circle in both degrees and radians for our reference. You may use this reference sheet on a quiz or a test. The trig unit circle is the image shown below, and it shows commonly used angles in trigonometry. 

Coterminal Angles

Coterminal angles are those whose measures differ by a multiple of 360 degrees. An angle can have both positive and negative coterminal angles. When finding the negative coterminal angles, we must rotate backwards on the coordinate plane, so make sure to draw your angles carefully!

Similar Triangles

Similar Triangles have the same angles (all 3) but different side lengths. We can solve for the missing side of a triangle by setting up a proportion and solving for the missing side. If we have a right triangle, remember that we can also use the Pythagorean Theorem to solve for a missing side.

In complicated diagrams, it often helps to redraw the similar triangles separately so that you can align your measurements properly. Sometimes you must look carefully to spot the similar triangles!

Geometry Review

We took a trip down memory lane and reviewed some important geometric ideas that might serve us useful this year in trig. Particularly, we looked at seven main ideas: Supplementary angles, complementary angles, congruent, vertical angles, corresponding angles, alternate interior angles, and alternate exterior angles. Then we did a few examples on what those problems could look like in the context of algebra.

Shout out to Cody Gracey for remembering most of this material from geometry, and shout out to 2nd period for being the most enthusiastic class that I've ever seen! WOW.

Pre-Test Day

We took the pre-test today since youtube.com is an epic fail on our internet. The first review sheet was given out in class and is due back on Thursday, August 29th for full credit. If you give it to me a day early, I will give you five extra bonus points.

When grading, I give two points per question - one for the correct answer and one for shown work.

If you were absent today and missed the pre-test, you have one week from today to make it up. If it's not made up by next Tuesday it's a zero - don't let that be you!! A great time to make up work would be during my service period (3rd), during my lunch (3rd lunch) or after school.

Shout out to Bre Boyd for making me laugh and to Brasha Campbell for reading out loud even though she had a sore throat today - love the perseverance!

Welcome to Algebra 3 aka Trigonometry! I am Ms. Doc and I will be your instructor this year. We are spending a large majority of the year in discussion of the six trigonometric functions (sine, cosine, tangent, cotangent, secant, and cosecant). I promise to make it manageable and a strong learning experience for all who take part in this class.

Today we went over the syllabus and filled out a brief survey so that I could learn more about you and how you've been taught in mathematics over the past year. We also talked about where students wanted to go to college, and WHY they wanted to be there (not just to study, but really, what DRIVES you to do what you do?)

Remember to bring back your signed syllabus! +5 bonus points if I get it tomorrow (Aug 20th), regular credit if I get it by Wednesday Aug 21st.  No credit if it's late.