# Engaging students: Graphing Sine and Cosine Functions

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Christian Oropeza. His topic, from Precalculus: graphing sine and cosine functions.

How could you as a teacher create an activity or project that involves your topic?

An activity for students to understand how to graph sine and cosine could consist the use of website desmos (Reference 1). In this activity students will be in pairs and must complete a worksheet that list different forms of the equation sine and cosine that illustrate some of the transformations for sine and cosine. One student will enter the equation onto desmos and the other student will draw the graph on a separate worksheet. The pair will switch roles after each equation, so both students understand how to interpret and draw a given sine or cosine equation. After all the equations have been graphed and sketched, each pair will move on to the next part of the activity in which they must manipulate the equation asin(bx+c)+d and acos(bx+c)+d on desmos where a,b,c, and d are all  numerical sliders that can be adjusted to help students visually interpret what transformation they represent. Finally, to prove that students understood the material, each pair will come up with a sketch of a transformation of sine or cosine and trade with another pair of students, in which they must figure out the corresponding equation that matches the given graph.

How can this topic be used in your students’ future courses in mathematics or science?

This topic comes up any subject that has sinusoidal waves, such as physics, calculus, and some engineering classes. For example, in calculus graphing the derivative of sine gives the graph of cosine. This shows students that the slope at any point on the sine curve is the cosine and the slope of any point on the cosine curve is the negative of the sine. The topic of sine and cosine is a crucial component  in electrical engineering (EE). For EE, there’s a class called, circuit analysis that has a section named “Euler’s Sine Wave” and “Euler’s Cosine Wave”, which incorporates the use of Euler’s formula (Reference 2). Also, in electrical engineering, there’s a machine called a “signal generator”, which sends different types of signals as inputs to circuit. This machine can alter the frequency and amplitude of the signal, where amplitude represents the amount of voltage inputted into the circuit. In math, there’s a topic called “Fourier Series” that also incorporates sine and cosine (Reference 3).

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.

Desmos (Reference 1), can be used to show students the different transformations of both functions. This way students can visually understand what each component is and how each component affects the functions y=asin(bx+c)+d and y=acos(bx+c)+d, where a is the amplitude, b is the period, c is the phase shift, and d is the vertical shift. Vision learning (Reference 4), is also a great website for students when they are introduced to the topic of sine and cosine. This website goes over the history of sine by relating it to waves and circles. The website first goes over how Hipparchus calculated the trigonometric ratios and how that led to the sine function. This website gives students a background on how the functions sine and cosine came to be over time. Also, this website talks about how when the Unit Circle is placed on a Cartesian graph, this illustrates how sine and cosine take over a periodic trend, so students can see why the graphs of sine and cosine are infinite if the domain is all real numbers.

References:

# Engaging students: Using right-triangle trigonometry

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Cameron Story. His topic, from Precalculus: using right-triangle trigonometry.

What interesting (i.e., uncontrived) word problems using this topic can your students do now?

Most right-angle trigonometry word problems involve giving two measurements of a triangle (angle, sides or both) and asking the students to solve for the missing piece. I argue that these problems are fine for practice, but one has to admit these problems encourage “plugging and chugging” along with their formula sheets.

To make things interesting, I would use something along the lines of this word problem from purplemath.com:

“You use a transit to measure the angle of the sun in the sky; the sun fills 34′ of arc. Assuming the sun is 92,919,800 miles away, find the diameter of the sun. Round your answer to the nearest mile,” (Stapel, 2018).

This is incredible! Using trigonometry, students can find out the diameter of the entire sun just by knowing how far away it is and how much of the sky the sun takes up. If you were to use this word problem in a experimental type of project, I strongly recommend using the moon for measurement instead; you can probably guess why measuring the sun in the sky is a BAD idea.

What are the contributions of various cultures to this topic?

One amazing culture to contribute to the study of triangles and trigonometry were the Ancient Babylonians, who lived in what is now Iraq about 4,000 years ago. Archaeologists have found clay tablets from 1800 BC where the Babylonians carved and recorded various formulas and geometric properties. There were several such tablets found to have been lists of Pythagorean triples, which are integer solutions to the famous equation $a^2+b^2=c^2$.

The Greeks, while going through their own philosophical and mathematical renaissance, gave the namesake for trigonometry. Melanie Palen, writer for the blog Owlcation, makes is very clear why trigonometry “… sounds triangle-y.”  The word trigonometry is derived from two Greek words – ‘trigonon’ which means ‘triangle’ and ‘metron’ meaning ‘measure.’ “Put together, the words mean “triangle measuring”” (Palen, 2018).

How can technology (YouTube) be used to effectively engage students with this topic?

In the YouTube video “Tattoos on Math” by the YouTube channel 3Blue1Brown (link: https://youtu.be/IxNb1WG_Ido), Grant Sanderson offers a unique perspective on the six main trigonometric functions. In the video. Grant explains how his friend Cam has the initials CSC, which is how we notationaly represent the cosecant function. Not only is this engaging because most students wouldn’t even think of seeing tattoos in math class, but also because Grant always backs up the mathematical content in his videos with beautiful animations.

Students know how sine and cosine functions are represented geometrically; these are just the “legs” of a right-angled triangle. Most students, however, only see the other four trigonometric functions as formulas to be solved. However, as Grant cleverly explains and visualizes in this video, all of these functions have geometric representations as well when paired with the unit circle. This video (moreover, this entire YouTube channel) can be helpful to those visual-learning students who need more than a formula to be convinced of something like the cosecant function.

References:

Palen, Melanie. “What Is Trigonometry? Description & History of Trig.” Owlcation, Owlcation, 25 July 2018, owlcation.com/stem/What-is-Trigonometry.

Stapel, Elizabeth. “Right-Triangle Word Problems.” Purplemath, 2018, http://www.purplemath.com/modules/rghtprob.html

# Engaging students: Defining sine, cosine and tangent in a right triangle

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Jessica Williams. Her topic, from Precalculus: defining sine, cosine and tangent in a right triangle.

How could you as a teacher create an activity or project that involves your topic?

I know of a good project/activity for the students to do that will be extremely engaging. You could either do this for an elaborate activity for your students or maybe an opening activity for day 2 of a lesson. For my class, I would get a square cookie cake, and have the slices cut into right triangles. I would allow each student to have a piece (but not eat it just yet). The students will be provided with rulers and a protractor. The students will each measure the hypotenuse of their cookie cake and the degree of whichever angle you would like them to measure, however each student should be measuring the same parts so do this unanimously). As a class, decide on an average for the measurements for everyone to use so that the data is not off. Then take the supplies away from the students and ask the students to find the rest of the missing sides and angles of their piece of cookie cake. They will also be provided with a worksheet to go along with this activity. This is a good review activity or al elaborate activity to allow further practice of real world application of right triangle trigonometry. Then go over as a class step by step how they solved for their missing angles and side lengths and make each group be accountable for sharing one of them. This allows the students to all be actively participating. Through out the lesson, make sure to tell the kids as long as they are all participating they will get to eat their slice when the lesson is done. Lastly, allow the students to eat their slice of cookie cake.

How does this topic extend what your students should have learned in previous courses?

Prior to learning about right triangle trigonometry the students will know how to use the Phythagorean Theorem to find how long the missing side length is of a right triangle. The students know basic triangle information such as, the sum of the angles in a triangle is 180 degrees. The students already know the difference between the hypotenuse and the other two legs. The students know that hypotenuse will be the longest leg and the leg across from the 90 degree angle. The students will also know the meaning of a fraction or ratio. The students may need some refreshing of memory on some parts of prior knowledge, but as teachers we know this is an extremely important part of a lesson plan. Even as teacher we tend to forget things and require a jog of memory. A simple activity such as headbands or a kahoot with vocabulary would be an excellent idea for accessing the students prior knowledge. This allows the students to formally assess themselves and where they stand with the knowledge. Also, it allows the teacher to formally assess the students and see what they remember or parts they are struggling on. This allows the teacher to know what things to spend more time on.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

Technology is always an amazing aspect of the classroom. Like stated above, a vocab review using headbands or kahoot would be a good idea for this type of lesson that DEFINITELY needs prior knowledge to be applied in order to succeed. Also, showing the students how to plug in sine, cosine, and tangent is crucial. They have seen these buttons on the calculator but they do not know what they mean or how to use them. Using an online TI on display for the class is great. I had to do this with my 10th grade students to make sure they understood how to use the 3 buttons. Also, when using arcsin, arccos, and arctan it can be confusing. Using technology to show the class as a whole is the best route to go. Also, technology can used as review for a homework assignment or even extra credit for the students. It benefits them by getting extra review and extra credit points. I found a website called http://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.SHAP&ID2=AB.MATH.JR.SHAP.TRI&lesson=html/object_interactives/trigonometry/use_it.html , which is a golf game that requires review of triangles and trigonometry. It allows the students to practice the ratios of SOH-CAH-TOA using a given triangle.

# Engaging students: Defining sine, cosine and tangent in a right triangle

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Jessica Williams. Her topic, from Geometry: defining sine, cosine and tangent in a right triangle.

How could you as a teacher create an activity or project that involves your topic?

I’ve actually had the opportunity to teach this lesson to my 10th graders last semester. It is a difficult concept for the students to understand, however if you teach it in a way the students are actively engaged, it helps extremely. Prior to this lesson, the students knew about the hypotenuse and knew the other sides lengths as “legs.” We started by calling 3 students up to the front to hold up our three triangle posters. (triangle cut outs with the 90 degree angle showing and then there was an agle missing). We asked the students how we could find a missing angle given only one side length. For starters, I demonstrated on one triangle by placing a spray water bottle at the missing angle given, and spray the water across.

What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

Trigonometry was originally developed for the use of sailing as a navigation method. The origins can be traced back to ancient Egypt, the Indus Valley, and Mesopotamia. This was over 4000 years ago. Measuring angles in degrees, minutes, and seconds comes from the Babylonian’s base 60 system of numbers. In 150 B.C.E, Hipparchus made a trigonometric table using sine to solve triangles. Later on, Ptolemy extended the trig calculations in 100 C.E. Also, in interesting fact is the ancient Sinhalese used trig to calculate for water flow. Persian mathematician Abul Wafa introduced the angle addition identities. As you can see, there are MANY different mathematicians who distributed to the topic of trigonometry. A lot of them built upon previous work and discovered new formulas, identities, etc. It’s amazing to see how even trigonometry is used to every day life. You always hear people say, “When will I ever use this is life?” and it bugs me to hear this. However, I always have examples of how math is used in our everyday world and from a past long ago that advanced us to where we are today.

How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

I would show this video that I found on youtube. I would exclude the first movie example that involves shooting, however the rest are great examples.

Showing movie clips to students is always a great way to grab their attention. Visually showing them that math is a part of movies, and every day life shows them that it is important. This video would also be great to use as practice problem, but blur out one of the side lengths or angles missing. You could play the movie scene then pause it on the part with the triangle and have the students solve for missing angle or side length. It would be a fun activity for the students and involve great practice. You could even make this a homework assignment. It’s engaging to watch and keeps the student’s attention while doing homework. The video shows that math is involved in dancing, buildings, etc. This activity also can excite students to try to find math in the movies or tv shows that the watch. You could assign the students to pay attention to to the next couple of shows or movies the watch and to bring back to class an example or two of how math was incorporated in it. Mathematics goes unnoticed because it is honestly part of our everyday norm/lifestyle.

http://www.newworldencyclopedia.org/entry/Trigonometry

# Engaging students: Graphing Sine and Cosine Functions

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Saundra Francis. Her topic, from Precalculus: graphing sine and cosine functions.

How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

To engage students you can have them record a song using the WavePad app. Have students choose a clip of their favorite song to record. Once they record the song they choose, the app will display the sound waves compiled that are used to create the sounds in the song the song. Students will realize that sound waves are in the form of the sine function. This will engage students since you would have related the topic of graphing sine and cosine functions to their favorite song. You could also have students create their own sounds and record them with the app to see the graph associated with the sound they made. Students can look at their sound and other classmates sounds are recognize differences in the waves, you can relate this to the equation f(x)=asin(bx+c)+d. You can them work with students to discover what the constants terms mean in relation to the parent function of sine.

How has this topic appeared in high culture (art, classical music, theater, etc.)?

Sine waves are the basis of sound. Have a piece by Beethoven playing while students are entering the classroom. Tell students that Beethoven was able to create music while he was in the process of becoming deaf. Ask students how they think Beethoven was able to create music in spite of that set back. After you have students share some answers show them the video above which explains how Beethoven’s music (all music) is related to sine waves. The music of his “Moonlight Sonata” is explained using math in the TED-Ed video. While Beethoven did not use this method to create his music he said that he knew what the music looked like. This will show students an example of how sine graphs are used in real life and get them interested in graphing sine and cosine functions.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.

Students will be given a TI-Nspire calculator in order to discover how changing the amplitude, period, horizontal shift, or vertical shift changes the equation of sine. Students will start with the graph of f(x)=sinx. They will then manipulate the graph on the calculator to change the function. Have them move the function up and down, right and left, and work with the slope of sinx and the slope of the x. Have students write dawn some of their new functions and sketch the graph. They will then compare how changing the graph effects the equation of f(x)=sinx. Introduce f(x)=asin(bx+c)+d . Give students some time to compare the functions that they created to the formula and describe how each constant changes the graph. Students will hopefully discover how the function f(x)=asin(bx+c)+d relates to amplitude, period, horizontal shift, and vertical shift.

# Borwein integrals

When teaching proofs, I always stress to my students that it’s not enough to do a few examples and then extrapolate, because it’s possible that the pattern might break down with a sufficiently large example. Here’s an example of this theme that I recently learned:

# Engaging students: Graphing the sine and cosine functions

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Jessica Bonney. Her topic, from Precalculus: graphing the sine and cosine functions.

How could you as a teacher create an activity or project that involves your topic?

A fun activity for students to learn how to graph the sine and cosine function would be having them build the graph using spaghetti and yarn. Students would start out with a simple warm-up to help them recall the different values of sine and cosine on the unit circle depending on the given angle. After the warm-up, I would then pair students off into groups of two and have them create the graphs, one creating the sine graph and the other creating the cosine graph. The first step in this activity would be for students to take their yarn and wrap it around the unit circle, marking each significant angle on the yarn with a marker. Next, students will create the x and y-axis on their paper, making the x-axis along the center of the paper (labeling it Θ) and the y-axis about 1/3 of the way from the left-end of the paper (labeling it either cosΘ or sinΘ). They then lay the yarn on the x-axis, with the end on the origin, which represents 0 radians, and using the marks they made on the yarn they will mark and label each point on the x-axis. Going back to the unit circle, students will then measure the major angles of either sine or cosine with spaghetti. This part is used to help solidify their understanding that the values of x and y correspond to cosine and sine. After measuring and cutting the spaghetti, students will then glue the spaghetti down to the matching angle on the coordinate plane. Once they have finished gluing their pasta down, students will take a marker and draw the curve. To end the lesson, I would have the students do a think-pair-share, answering the following question: Why is the function curve wider that the unit circle? After, I would have students compare their graphs and demonstrate how they found their graph.

How can this topic be used in your students’ future courses in mathematics or science?

Graphing the sine and cosine functions is a topic that students will carry on with them throughout the rest of their future science and mathematics courses. For starters, they will need to know how to do this for all advanced calculus or trigonometry classes they will take in high school or even in college. An example of this would be, when the students learn how to derive the tangent, cotangent, secant, and cosecant functions and graphs. Next, students will use this more in depth in their future physics courses. They will be able to relate waveforms and vibrations to that of specific sine and cosine graphs. Vibrations are graphs with the equations y=sin(t) or y=cos(t), and the time needed for one oscillation across the x-axis is referred to as a period. Waveforms are graphs with the equations y=sin(x) or y=cos(x), and the distance needed for one oscillation across the x-axis is referred to as a wavelength. As you can see, this particular topic in pre-calculus is an important piece in laying the foundation in their future academics and beyond.

What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

For starters, the word trigonometry comes from the Greek word trignon, meaning “triangle”, and metron, meaning “to measure.” Before the 16th century, trigonometry was mainly used for computing the unaccounted for parts of a triangle when the other parts were given. When it comes to ancient civilizations, Egyptians had a collection of 84 algebra, arithmetic, and geometry problems called the Rhind Papyrus. This showed that the Egyptians had some knowledge about the triangle, almost like a “pre-trigonometry”. It wasn’t until the Greeks, that trigonometry began to make sense. Hipparchus was the first to construct a table of the values of trigonometric functions. The next key contribution to trigonometry as we know it came from India. The author of the Aryabhatiya used words for “chord” and “half-chord” which was later shortend to jya or jiva. Following this, Muslim scholars translated the words into Arabic, which was then translated into Latin. An English minister, Edmund Gunter, first used the shortened term that we know, sin, in 1624. In 1614, John Napier invented logarithms, the final major contribution of classical trigonometry.

References:

https://www.britannica.com/topic/trigonometry

http://betterlesson.com/lesson/437440/graphs-of-sine-and-cosine

http://www.algebralab.org/lessons/lesson.aspx?file=trigonometry_trigperiodfreq.xml

# Engaging students: Deriving the double angle formulas for sine, cosine, and tangent

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Daniel Adkins. His topic, from Precalculus: deriving the double angle formulas for sine, cosine, and tangent.

How does this topic extend what your students should have already learned?

A major factor that simplifies deriving the double angle formulas is recalling the trigonometric identities that help students “skip steps.” This is true especially for the Sum formulas, so a brief review of these formulas in any fashion would help students possibly derive the equations on their own in some cases. Listed below are the formulas that can lead directly to the double angle formulas.

A list of the formulas that students can benefit from recalling:

• Sum Formulas:
• sin(a+b) = sin(a)cos(b) + cos(a)sin(b)
• cos(a+b) = cos(a)cos(b) – sin(a)sin(b)
• tan(a+b) = [tan(a) +tan(b)] / [1-tan(a)tan(b)]

• Pythagorean Identity:
• Sin2 (a) + Cos2(a) = 1

This leads to the next topic, an activity for students to attempt the equation on their own.

How could you as a teacher create an activity or project that involves your topic?

I’m a firm believer that the more often a student can learn something of their own accord, the better off they are. Providing the skeletal structure of the proofs for the double angle formulas of sine, cosine, and tangent might be enough to help students reach the formulas themselves. The major benefit of this is that, even though these are simple proofs, they have a lot of variance on how they may be presented to students and how “hands on” the activity can be.

I have an example worksheet demonstrating this with the first two double angle formulas attached below. This is in extremely hands on format that can be given to students with the formulas needed in the top right corner and the general position where these should be inserted. If needed the instructor could take this a step further and have the different Pythagorean Identities already listed out (I.e. Cos2(a) = 1 – Sin2(a), Sin2(a) = 1 – Cos2(a)) to emphasize that different formats could be needed. This is an extreme that wouldn’t take students any time to reach the conclusions desired. Of course a lot of this information could be dropped to increase the effort needed to reach the conclusion.

A major benefit with this also is that even though they’re simple, students will still feel extremely rewarded from succeeding on this paper on their own, and thus would be more intrinsically motivated towards learning trig identities.

How can Technology be used to effectively engage students with this topic?

When it comes to technology in the classroom, I tend to lean more on the careful side. I know me as a person/instructor, and I know I can get carried away and make a mess of things because there was so much excitement over a new toy to play with. I also know that the technology can often detract from the actual math itself, but when it comes to trigonometry, and basically any form of geometric mathematics, it’s absolutely necessary to have a visual aid, and this is where technology excels.

The Wolfram Company has provided hundreds of widgets for this exact purpose, and below, you’ll find one attached that demonstrates that sin(2a) appears to be equal to its identity 2cos(a)sin(a). This is clearly not a rigorous proof, but it will help students visualize how these formulas interact with each other and how they may be similar. The fact that it isn’t rigorous may even convince students to try to debunk it. If you can make a student just irritated enough that they spend a few minutes trying to find a way to show you that you’re wrong, then you’ve done your job in that you’ve convinced them to try mathematics for a purpose.

After all, at the end of the day, it doesn’t matter how you begin your classroom, or how you engage your students, what matters is that they are engaged, and are willing to learn.

Wolfram does have a free cdf reader for its demonstrations on this website: http://demonstrations.wolfram.com/AVisualProofOfTheDoubleAngleFormulaForSine/

References