Episode 035: sohcahtoa example #1- 3-4-5 triangle angle calculation.

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One of the best ways to get comfortable with step-by-step processes is to see them in action. The first example problem we'll work through deals with calculating a specified angle of a 3-4-5 right triangle. This triangle is well understood and you may even know the answer we're solving for before we get started. A quick google search will tell you the answer. These types of well-known problems are great to try, because you can be sure that your answer is right.

Q: A right triangle has sides of lengths 3 and 4. Calculate the non-right angle closest to the side of length 4.

So, let's roll through the steps one by one and see how far we can get.

1) Identify the right angle in your triangle. This confirms that you will be able to use the sohcahtoa rule.

This will eventually become something that you can maybe even do in your head, but for now, let's be exhaustively descriptive. We've circled the right angle in red and added a little square, which is the common notation for a right angle. Check. Done. Next step.

2) Identify your angle of interest. This will change which sides are opposite and adjacent.

The identified angle is circled in red. Now we know which angle is the hypotenuse and can now identify the opposite and adjacent sides for this specific problem.

3) Identify your opposite and adjacent legs of the triangle.

Especially at the beginning stages, explicitly marking each of these angles is very important. It can lead to mistakes in the coming steps if you aren't tiresomely descriptive of what is what in this step.

4) Using the sohcahtoa rule, set up a relationship using sin/cos/tan, the angle, and two of the adjacent/opposite/hypotenuse sides.

The biggest trick here is to sort of "understand what you have". In this problem, we have the opposite and adjacent sides. We're looking for the angle, theta. The quickest way to our goal is to use toa, or tan[theta] = opp/adj. In this equation we have three variables, theta, adj, and opp. Two of them we already know. The reason we didn't choose sine or cosine is because those equations require knowing the hypotenuse, which isn't given in the problem. We could calculate it easily enough, but it's extra work which is not required of us. Good, concise mathematics is born out of a deep, internal laziness embedded in any good physicist or mathmetician.

5) Solve the equation for your desired variable.

At this point, we simply need to calculate the arctangent of our fraction, 3/4.

6) Box your answer, you're done!

At last, we arrive at our answer, 36.86 degrees. Make sure if your calculator gave you something like 0.64 that you understand that this is actually the right answer, but that the calculator is in radians mode, not degrees!

Episode 034: 6 Steps to Applying sohcahtoa to any Right Triangle Problem!

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One of the easiest ways to work problems is to break it into nice, digestable chunks. If you can identify a step-by-step process to do each chunk, you're good to go. Here are six steps for applying the sohcahtoa trick to right triangle problems.

1) Identify the right angle in your triangle. This confirms that you will be able to use the sohcahtoa rule.

2) Identify your angle of interest. This will change which sides are opposite and adjacent.

3) Identify your opposite and adjacent legs of the triangle.

4) Using the sohcahtoa rule, set up a relationship using sin/cos/tan, the angle, and two of the adjacent/opposite/hypotenuse sides.

5) Solve the equation for your desired variable.

6) Box your answer, you're done!

Episode 033: Why does your calculator give wrong values for sine, cosine, and tangent? Avoid frustration with radians and degrees mode!

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As a young college student, we were one of the first classes to do "online" physics problems as homework. The idea was that at the beginning of each week, the professor would hand out 20-25 problems on a sheet of paper. They were all related to the topics being covered during the week. The set was due next Monday. However, these were not to be turned in on paper (this used to be how things were done). Rather, each value was to be input into a customized website. Each student had different numerical values for each problem, so no two sets were alike. This eliminated the ability for students to copy each other outright, which I must admit was a rather nice feature of my high school life.

Many problems in first-year physics involve solving for an angle, either as the final answer, or as some sort of mid-way answer in the problem. This was death for many of my comrades. It was also death for me a few times, although I never had a major problem with it. To solve for an angle, you usually have to use an inverse-trig function. This is essentially a function which "undoes" sine, cosine, or tangent. They are referred to as arcsine, arccosine, and arctangent. So, if you know that the sine of theta is equal to 0.5, then the arcsine of 0.5 gives you the value of the angle.

You have to be certain that the value of this angle makes sense. The advent of the calculator did away with huge sine/cosine tables, but it also automated something that beginner students may not have understood to begin with. That is, the calculator can do trigonometry in two different "modes": radians and degrees.

Somewhere on your calculator there is a wonderful button which toggles between these two modes, and somewhere on your screen there is an indicator of "rad" or "deg". This means that for all sine, cosine, and tangent functions, it is expecting an angle in radians if "rad" is selected, and "degrees" if degrees if "deg" is selected.

If you're not familiar with radians and degrees, I encourage you to take a listen to Episode 011: Radians to Degrees in 2 Steps.

The best piece of advice I can give is when doing any trig function, think about your answer and see if it makes sense. Knowing the angle, you should be able to have a rough idea of your sin/cos value. I encourage you to check out Episode 024: Sines and Cosines by Counting to 4! for more information about guesstimating the value. If what you're seeing on your calculator is way off from this, you're probably in the wrong mode.

When using arcsine, arccosine, arctangent. Think about this value the calculator is showing you. It's supposed to be an angle. 2 pi is somewhere in the range of 6.25. If you're looking for a value in degrees and you have an angle less than 6, either your angle is REALLY small, which 1 in 15 times it is, or you're in the wrong mode. Usually the answer is less than 90 degrees, which is less than 1.5 radians. If the angle is less than 1.5, chances are you're in the wrong mode.

We used to spend a long time going through problems to find out where my friends and sometimes yours truly veered. This was many times just that place. Typing in an angle of 0.2435. The computer doesn't know you're in radians! This brings to mind a wise aphorism from physics teachers across the board: Check Your Units!

For some reason a lot of people don't associate degrees and radians with units. It's a very small detail to lose in the grand scheme, but keeping an eye out for it can save you 20 points on the big test. Don't let them slip through your fingers.

Episode 032: sohcahtoa 5 question pop quiz!

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Test your knowledge.

1) What is sohcahtoa?

2) What do the S, C, T, A, O, and H stand for in sohcahtoa?

3) In a right triangle, with respect to one non-right angle theta, the side which forms the angle, alongside the hypotenuse, is referred to as what?

4) What is the other side of the triangle referred to as?

5) What are the values of sine, cosine, and tangent, defined in terms of sides of a right triangle?

Tune into the episode to find out the answers!

Episode 031: sohcahtoa, sine, cosine, tangent, and the right triangle.

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"Sohcahtoa. My Native American friend." These were the words with which the acronym sohcahtoa were first conveyed to me by my high school teacher. "Racism" notwithstanding, I'll never forget it. And it's wildly useful to remember the relationship between sides of a right triangle and the sine, cosine, and tangent of an angle.

You will use these relationships a million times or more in the first two years of physics. This topic will span several episodes, but right now we will concentrate on introducing the sohcahtoa moniker and review some facts about triangles. I highly recommend checking out Episode 027: 27 Triangular Facts and Factoids You Must Know! as a prereq to this, in the event that you're not familiar.

The elements of sohcahtoa, each letter in sohchatoa stands for one of the following:

Sine

Cosine

Tangent

Adjacent

Opposite

Hypotenuse

Of course, sine, cosine and tangent are trigonometric functions and adjacent, opposite, and hypotenuse refer to sides of a right triangle. With this in mind, given any right triangle, we can draw the following three relations of any angle, theta:

In a right triangle, the sides of the triangle can be specified with respect to one of two non-right angles, theta. The hypotenuse is always the longest side, and is the only side which does not form the right angle.

Given an angle theta, the adjacent side is always the side which forms the angle theta with the hypotenuse.

The remaining side, which does not form the angle theta is referred to as the opposite side.

For the other angle, the opposite and adjacent sides are reversed.

Knowing these relationships is key to doing vector operations in your first year. Next, we will outline the steps to do any sohcahtoa problem, as well as run through some basic examples.

Episode 030: Wardenclyffe Tower, pt. 2

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This trip has been 3 years in the making. Nikola Tesla. Mad Scientist. Master of Electricity. The guy who has pictures of him both HOLDING an illuminated lightbulb, and sitting reading the paper between two tesla coils with electricity wiring out every which way. He was doing things a hundred years ago that top electrical engineers of today still scratch their heads over. And he was doing some of it within a 20 mile radius of where I live. And I have lived here for OVER 3 years and still haven't taken the 20 minute drive out to this monumentally historic location. I wonder if he did any of those insanely cool photo ops at Wardenclyffe.

I wonder how many people knew what WiFi was even 15 years ago. The idea of having wireless internet hotspots in 2000 was crazy. Wirelessly transmit information from the kitchen to your bedroom!?!? This guy wanted to do it from Long Island to the United Kingdom. And he wanted to do it ONE HUNDRED YEARS AGO.

Is it ironic that I put in my request for directions to Wardenclyffe Tower into Google Maps, which then used all kinds of wireless technology to plot out my route? There were a few of those little ironies on this journey. Google Maps was also able to map it out as Wardenclyffe Tower. There are some really great things about being alive in the modern day.

On my way out, I started thinking about the man himself, and what it must have been like to live out here a hundred years ago. My father-in-law says that it's not even the same place from thirty or forty years ago, where this far out east was pretty much farmland and woods. Now it's wildly overpopulated and seriously suburban. I wonder what Tesla's ride out to Wardenclyffe Towers must have been like. My musing was interrupted by Google Maps warning. We were getting pretty close.

It didn't look terribly impressive. The ride up that is. I'm not sure what I was expecting. I guess I was expecting a lot of forest, and ominous approach. What I got was driving down 25A, near Rocky Point/Shoreham, a lot of suburban houses, a firehouse, and a lot of strip malls. A small taste of what someone wanted to do to Wardenclyffe Tower. (see below in all it's "unadultered glory")

Nothing against the fine folks who run those businesses, of course. Everyone's got to make a living. And this far out east, they do it in a really classy way. They keep all of the strip malls, Subways, Walgreens, gas stations, etc. within these really tight restrictions to make the whole town look a lot nicer, to make everything look faux-historical. It does give the whole homogenization of community a much softer feeling. But I think most of us feel like maybe we could forego another round of it to preserve Mr. Tesla's legacy. Then I was finally there. It comes up very anticlimactically, it's across the street from a fire department. Next thing you know, I'm on Tesla St.

Tesla St. is actually a residential street from the looks of it. There's houses on it on the right side, and on the left, barbed wire fence protecting Wardenclyffe Tower. Then I get my first look at the facility itself. It is seriously cool and historic looking. It also has a monster chimney coming out of the top, which makes it not entirely hard to envisage where a 187-foot tower must have loomed a century earlier.

It's at this point that I start getting those spine tingles. It starts to hit me that Nikola Tesla himself had walked not far from where I was standing. He walked through the gates and into that building to do serious work. It's a really crazy vibe that I think most people can relate to. I think it has something to do with the mixing of stories that you have been told from early ages and the collision of that fantasy with standing somewhere where those things actually happened. It's a wonderful feeling that I haven't felt enough times in my life.

From this street I'm able to make it over to the side of the building. The people who are going to clean this up into a museum have a lot of work ahead of them. There's a lot of graffiti on the side of the building. On the one hand I want to stand up for Mr. Tesla's property, on the other hand, if I was a graffiti artist, this has to be a holy grail type place to lay your tag.

I then make my way to the main entrance, which is somewhat further west of Tesla St. Everything is of course closed, but I am able to see the new Tesla statue that has been put in place. It looks awesome. Apparently the President of Serbia, Tomislav Nikolic, came out to personally dedicate the statue. They were looking for a site worthy of its dedication. Wardenclyffe fits the bill, no question. Engraved on the statue is his equation for magnetic flux density. In yellow or gold. It looks awesome.

I drove around the other side of the facility. Apparently it's like 16 acres of land. It seems very small, and it's totally surrounded by single family homes. Around the back of the land are huge power lines which carry multi phase alternating current which Tesla helped design, through the backyard of the Wardenclyffe Tower facility, to god knows where. Another one of those fitting ironies in this journey. I wonder how many people who live around here know what sacred ground is just around the corner.

Episode 029: Wardenclyffe Tower, pt. 1

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Wardenclyffe Tower. What a vision. In 1901 Nikola Tesla set out to build a wireless communications station on Long Island, NY to communicate with the UK. It was to be 187 feet tall. A real monster. It would be able to send messages, facsimile, and telephonic communications. By 1907 Tesla was asking for more funding. JP Morgan said no. In 1917 the tower was pulled down and sold for scrap to help recoup some of the mounting debt Tesla was amassing.

The land was sold to a photography company and was used to manufacture photographic components for 50 years. But the main building remained intact.

A few years ago the land went up for sale. There were to major bidders: a not-for-profit who wanted to buy the land and turn it into a museum dedicated to Nikola Tesla, and a business who wanted to level the land and turn it into a strip mall.

The Oatmeal, (for his side of the story click here) a classy web comic found out about this and helped expose a crowd-funding campaign to raise the money to preserve this land as a historical site. The entire effort was a success and soon we will have our own Tesla Museum!

I was following this story a few years ago when it was fresh. I realized how close I was to such a great historical landmark and set out to travel there one weekend. Alas, life got in the way. I came across this story again a few weeks ago and got super excited. Elon Musk, Tesla Motors CEO, donated $1 million to the cause as well as promised to put a tesla motors charging station in the parking lot. What a classy guy.

As part of my dream to own a Tesla Motors car and my general obsession with N. Tesla in general, I am now renewing my committment to have a drive out to Shoreham NY, and stand at the fence, and wax poetic, and drool. More soon on Wardenclyffe Tower, pt. 2!

Episode 028: Staying in the Zone and Its Effects on your Semester!

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In my podcast I always end with something akin to "Stay frosty, stay in the zone, and I'll catch you all on the flip side, take care." Sometimes I wonder if the weight of those words is lost on listeners as just a long-winded "adieu". But there is something so powerful in those words that I thought it was worth expounding upon in some detail.

Without going into too much detail, I am currently in the process of doing something that I don't want to be doing. My first reaction is to just get wildly frustrated and spend most of my available energy seething and focusing on how I don't want to be doing this particular thing. Once I am able to let go of this feeling, I am able to get things done quickly and efficiently. I think the effect being fired up and having the right mindset can change the outcome of your performance in physics or anything else profoundly.

If you are a physics major, or considering dedicating your life to physics in some capacity, here is something you need to consider: if you don't love this, you probably shouldn't do it. If you're not interested enough in physics to talk about it casually, this is not the craft for you. If watching science shows on TV doesn't keep you from flipping the stations, consider a new course now.

On the other hand, if reading science magazines does get you all hot and bothered, this still may not be the path for you. But it is a sign you are on the right track. There are certain courses after the entry level stuff called screen courses. They are unbearable. That is the purpose of them. They are unnecessarily time consuming and difficult. The idea is that anyone who doesn't really have a fire for this line of work will not make it through the screen. If you can't let the material consume you and embody almost every waking moment, how will you be able to do it ad infinitum for the rest of your life? However, if this is truly for you, don't lose hope. It's ok to do terrible. You will make it through.

What will help is getting fired up and being in the zone. It can magnify your focus and stimulate your curiosity to go down dark alleys on homework assignments, and not get frustrated when the problem isn't working out. Getting finished isn't the goal. Understanding what's going on is the whole purpose. The idea is to ram the material so far into your brain that the topics are no longer forebrain fuzzy ideas. They are embedded in the fabric of your consciousness. You can talk comfortably about it with professors and other classmates because you aren't constantly re-evaluating yourself on the fly. You know this stuff.

If the material is still in your frontal lobes and not in your fish brain by the time the exams come up you are not in the position you should be in for that test. Exams that I did phenomenally well on I don't remember the majority of the test period. When I was confident I could go into auto-pilot mode and let my subconscious do the work. This is how I do my best programming. I feel like I go into a sort of coma while it's going on. I kind of just stare at the screen, or homework paper blankly and let the information flow from my fingers. This is being in the zone. This is having an understanding of what you're doing. If you have to think about it, you don't know it very well.

The time between exams is for anxiety and worrying about the next exam. It's about keeping the material in your head and letting it sink deeper and deeper into the sands of your subconscious. Only when it's buried so deep that you don't even really know it's there will you be truly prepared to move forward.