Transcription
Sine, cosine, and tangent. We use these functions in a trigonometry class. As well as their friends, secant, cosecant, and cotangent. They also come along from taking the reciprocals. We use them in solving math problems involving oscillatory behavior, things involving right triangles, whether that be studying waves, finding side lengths, or finding angles.
When we construct our vanilla trig functions from the standard unit circle, we get this picture with the X values representing the cosine and sine being represented by the Y values. Today, I will show you some evil or lesser-known trig functions and where they are used. So, here's the list.
We begin with historical navigation trig functions, the versine and haversine. It's one of the original trig functions appearing in early trigonometric tables. Looking at our common trig functions, it is defined to be 1 - cosine theta, which is also equal to this. We also have vercosine or versine. There is also coversine or in shorthand, the cover sine.
If we look at this geometrically on a unit circle, we see cosine and sine, but this little extension of the remaining radius past the cosine that touches the end of the circle is what versine is. This figure is also called a sagitta, which is Latin for arrow. If the arc ADB of the double angle delta equals 2 theta is viewed as a bow of the chord AB, this chord extends the sine line and represents the string of an arrow. This means that versine is essentially the arrow shaft. I guess ancient mathematicians of yesteryear were just as creative as the ones we have today.
A practical reason for versine existing [music] is that it's always positive, and since it's positive, you can apply logarithms to it. This is valid except for the endpoints of 0 and 2 pi n. This is where versine is equal to 0, and at those points, the logarithm is not defined. I'm not sure why you want a log of versine or if it has any practical uses, but you can definitely do this.
We talked about versine, and now here is haversine. This was especially practical for sailors of older times. Haversine is by definition just half of versine. But, once we have this definition, it has a much deeper use in something called the haversine formula. Suppose you are the captain of a ship, and you don't know where the hell you're going. And you want to know the distance between the two points on a spherical Earth that follows along a great circle. Now, you just can't use a really big tape measure, as you have to go through water. So, the mathematician comes in to help. You know the latitude and longitudes. With this information, you're able to apply the haversine formula to help you find a distance. It's a bit messy, but it accurately provide the answer of a great circle arc length. In our case, the distance.
This is how the formula works. We begin with haversine theta, which is equal to haversine delta psi plus cosine psi one, cosine psi two, haversine delta lambda. Where psi one and psi two are latitude points, and lambda one, lambda two are longitude points. And then we have delta psi, which is equal to the difference between the latitude endpoints, and delta lambda, which is equal to the difference between the longitude endpoints. The haversine function computes half of versine of the angle theta. And by the properties from before, haversine is also equal to sine squared of theta over two, which is also equal to 1 minus cosine over two. To solve for theta from the distances between the latitude and longitude explicitly, we have theta is equal to two arc sine of the square root of haversine. From this, we can use the arc length formula of a circle, which is S equals R theta, with theta being in radians, and R being the radius of the Earth.
Let's see an example of this. Let's find the distance between New York and Los Angeles. New York is 40.713° North in latitude of -74.006° West. Los Angeles is 34.055° North and -118.243° West. Now, we will apply the haversine formula to determine the distance between these two cities on the globe.
>> All right, Stewie. So, you want to know how far New York is from Los Angeles. It's It's really far, Stewie. Like really, really far. Like that time I walked to the fridge and it turned out we were out of Pawtucket Patriot Ale. Truck.
>> Oh, for the love of yes, Peter, it's [music] far. Now, shut up and listen because I'm only explaining this once. First, you need the coordinates. New York is at 40.71° North, 74.01° West. Los Angeles is at 34.05° North, 118.24° West.
>> [laughter]
>> Degrees, like a fever. I once had 104° and Lois made me sleep on the couch.
>> [laughter]
>> That's That's not relevant. Now, you subtract the latitudes. 34.05 - 40.71 gives you -6.66 degrees. That's your delta phi. Write it down, you fat imbecile.
>> Oh, oh, oh, oh, I know delta. That's an airline. Me and Quagmire flew delta to Vegas that one time and giggity
>> Peter, focus. Now, subtract the longitudes. -118.24 - -74.01 gives you -44.24 degrees. That's delta lambda.
>> Lambda? He, he, he, he. That sounds like a frat. Lambda, lambda, lambda. Those guys were nerds, Stewie.
>> I am a nerd, you insufferable oaf, and I will destroy you. Now, you plug everything into the haversine formula. Take 1 - cosine of delta phi + cosine of 40.71 * cosine of 34.05 * the quantity 1 - cosine of delta lambda, then divide the whole thing by two.
>> Oh, yeah, yeah, yeah. So, it's like a recipe, like for a pie.
>> It is actually somewhat like a recipe, yes.
>> Oh, and see, I'm smart.
>> You are categorically not. Anyway, [music] you get haversine theta equals 0.0926. Then you take the square root, giving you 0.3043. Then arcsin of that, giving you 17.72 degrees. Double it, you get 35.44 degrees total central angle.
>> Central angle? Like the middle of a pizza slice? I like pizza.
>> Convert that to radians by multiplying by pi over 180. You get 0.6188 radians. Then multiply by Earth's radius, 6,371 km.
>> How big is that in Pawtucket city blocks?
>> It is irrelevant. The answer is approximately 3,942 km, or about 2,449 miles, which is how far I wish you were from me at all times.
>> Wow, that's pretty far.
>> Yes, I'm going to my room.
Next up in the class of these trig functions, we have the external secant. It really is just a name change for secant minus one. That is it. That is what external secant is. The external secant becomes practical when it comes to civil engineering and railroad surveying to calculate the external portion of a secant line. First introduced in 1855 by American civil engineer Charles Haswell. He used it along the versine function for designing and measuring circular sections of railroads. [music] Into the 20th century, when the US was building roadways, external secant was once used again to figure out the bends.
Let's look at what secant really means. The word secant comes from Latin meaning to cut. Very much like a secant line cutting through a circle and intersecting it twice. External secant was primarily used by a railroad surveyor or a civil engineer as when they are designing bends, circular arcs, or simple curves were the main method of construction. To determine such arcs, many of those working in the field had to do repetitive trigonometric calculations to measure and plan out circular sections of a track. These were done in a time before computers. So, many of these calculations were done by hand. This required the mathematical Bible of the time, table of logarithms and trigonometric values. Logarithms have the interesting properties of converting multiplication to addition and using logarithmic tables of trig tables further save numerical labor by reducing the number of necessary table lookups.
The external secant or external distance of a curved track is the shortest distance between the track and the intersection of the tangent lines from the ends of the arc. If we look back to versine, the versed sine of a curved track distance is the furthest distance away from the long chord. Using logarithms, Haslett found that looking up the log of an exsecant and versine provided more accurate results than calculating the same quantity from trigonometric tables. By 1913, Haslett's approach would be later widely adopted by American railroad industry that tables of external secants and versines were more common than tables of secants. Into the 20th century, these values were still used when railroads started building more sophisticated arcs. Sophisticated arcs included the Euler spiral, which is a track transition curve between straight or circular sections of different curvature. These spiral curves were approximating using exsecants [music] and versines.
We have sinh, cosh, and tanh. These are analogous to sine, cosine, and tangent, with the only difference being that the standard trig functions of sine, cosine, and tangent [music] are based off of a unit circle, x² + y² = 1. The hyperbolic trig functions, sinh, cosh, and tanh, are based off the unit hyperbola, x² y² = 1. This is how sinh is defined. And this is how cosh is defined. And this is how tanh is defined. [music]
Of all the trig relatives we will talk about today, the hyperbolic trig functions are the ones that have the most applications and show up in many places of mathematics and physics. One cool application comes from space-time and relativity. Here's something that I once had on my linear algebra midterm. The problem was mainly about multiplying matrices, but the professor provided a bit of a twist on the problem. He provided an intimidating problem for second-year linear algebra students. He made this mess to mainly intimidate students into doing a lengthy and most likely error-prone matrix multiplication. Essentially, he wanted us to find A transpose A. Remembering that matrix multiplication is not commutative, this will take some time, especially as a single problem on an hour-long midterm.
Looking at the matrix, we can see how it represents a transformation. The top left and bottom right are gamma. The faster you go, the bigger this gets, and the more space-time gets distorted. When your velocity is zero, it [music] equals one, and nothing else happens. The bottom left mixes velocity into the space coordinate. It's saying how far has a moving frame traveled. The faster the frame moves, the bigger velocity V, the more the position shifts. The minus sign in front means [music] that the moving frame sees you going the other way. The top right is a mix of positions into the time coordinate. The C squared in the denominator makes it tiny at non-relativistic speeds. It's essentially saying where you are in space affects the time you measure. This Lorentz transformation matrix leaves the space-time interval unchanged. We are able to do the physics and use this chain of logic. First, the speed of light is constant for all observers. This forces the space-time metric to have a minus sign. C squared dt squared minus dx squared. That minus sign makes the geometry hyperbolic and not Euclidean. Transformations that preserve hyperbolic geometry trace out a squared minus b squared equals 1. And that's simply the definition of cosh and sinh. As someone who knew these hyperbolic trig functions, I substituted cosh psi equals gamma and sinh psi equals gamma beta. With beta equals v over c, representing velocity as a fraction of the speed of light. While everyone else in the lecture hall was multiplying and messing with radicals and simplifying terms, I did this rapid substitution and was able to get the metric's product in a minute. Doing this, the easy part was the hyperbolic Pythagorean identity. Where since these functions are based on the unit hyperbola, cosh squared minus sinh squared equals 1 and sinh squared minus cosh squared equals -1.
Here's another fun thing about sinches and coshes. As a calculus teacher, one of my favorite lessons is teaching about area and arc length through integration. And there is a question I ask my students at times. Is there a function such that area under the curve equals arc length? Answer is yes, and that function is the cosh function. The derivative of cosh is sinh. It does not have a minus sign. This comes from its exponential definition. Take the integral from 0 to x of cosh t dt and we get sinh x.
>> We take the arc length integral of the cosh curve and I use the arc length formula. The derivative of cosh is sinh. So, placing it under the radical, we get square root of 1 + sinh squared x, which by our hyperbolic trigonometric identity, we get cosh. So, ultimately, these integrals are the same.
Now, this is a cool one. The Gudermannian function relates a hyperbolic angle, say psi, to a circular angle, phi. This is denoted as gd psi. This one function bridges regular trig functions with hyperbolic trig. Some history about it is that it was introduced in 1760 by Lambert and then later named after Gudermann. Don't know why he took the credit. The Gudermannian function and its inverse became especially helpful in finding values of hyperbolic functions given only a table of circular functions. This is the setup. Given a standard unit circle and a unit hyperbola, we have the graphs of x squared + y squared = 1 and x squared - y squared = 1. With this, we can see how the Gudermannian function relates area of a circular sector to the area of a hyperbolic sector. And when we talk about areas, one thing immediate pops up, the integral. That being said, the real part of the Gudermannian function is defined for psi, where it is the integral of the hyperbolic secant. Similarly, we have the inverse if you want to go the the way around.
Our next bunch of trig functions are the sine and cosine integrals. These are used in antenna theory and Gibbs phenomenon. We have the sinc function, which is a sampling function. It is at the heart of digital signal processing and image reconstruction. We also have Fresnel integrals, which are crucial in describing diffraction patterns when light passes through an edge.
We begin with sinc x. This we probably seen before. It is sin x over x. In a standard calculus one class, you may have learned that the limit as x tends to zero of this function is one. The term sinc comes from the contraction of the function's full Latin name, sinus cardinalis, or cardinal sine. It is a shorthand for the sin x over x function that occurs so often on Fourier series, it merits its own notation. When you try to integrate the sinc function, we are left with this non-elementary integral. Now, the one with a capital S looks like this. And the one with a lowercase s looks like this. Plotting this function, we get this type of behavior and an asymptote. Similarly, when you take the integral from zero to x of 1 - cos t over t dt, this is known as cin x. Kin x?
Next up, we have the Fresnel integrals and their auxiliary functions, which are defined as the following. S of x is the integral of sin t squared dt. C of x is the integral of cos t squared dt. These are its auxiliary functions. Cool thing happens when we take the parametric curve of a Fresnel sine and cosine integral, S of T, C of T. This gives us an Euler spiral. Fresnel integrals are tricky to solve without knowing a few special integration techniques. We won't cover them here, but with the use of modern computers we can get a numerical approximation. A closed-form solution of these Fresnel integrals are not available in terms of elementary functions. Fresnel integrals are used in electricity and magnetism. They are used in the calculation of finding electromagnetic field intensity in an environment where light bends around opaque objects.
Quadrants [music] and spread. Instead of measuring lengths and taking square roots, quadrants calculates the squared Euclidean distance between two points. This makes calculations entirely algebraic and avoids nasty irrational numbers. Rational trigonometry avoids dealing with diagonal lines or measuring an angle. When we do that in classical trigonometry, we'll end up with things such as the square root of two or square root of three and transcendental numbers such as pi when dealing with angles. When we do these, we will never get a clear and definite solution.
>> [music]
>> The irrational numbers will show up and this would require using numerical methods or infinite series expansion to calculate basic geometry. In the early 2000s, Wildberger decided to create an entirely new type of trigonometry to fix this issue. And that new type of trigonometry is rational trigonometry. His goal was to rebuild geometry only using addition, subtraction, multiplication, and division. Wildberger threw out the concepts of Euclidean distances and angles, which are completely arbitrary, and replaced them with things called quadrants and spread.
In classical geometry, we measure the separation between two points through Euclidean distance. But, this distance comes from Pythagorean theorem and thus requires a square root. And that leaves some numbers expressed irrational such as a square root of two. The distance formula requires a square root, which is the root of the problem. Quadrants is distance squared. That is when we have classical distance, it is a square root of the difference between the points. But, rational quadrants does not have that square root. Sticking to quadrants allows us to stick only with rational numbers. This will make Pythagoras really proud and probably not lose his Angles are dependent on the measurement of circles,
>> [music]
>> in particular arc length. To find angles, we use radians, which are by default defined by pi. We also need to use sines and cosines, all of which are notoriously difficult to calculate without the use of infinite series. Spread replaces the concept of an angle. It is a dimensionless ratio that measures how opened up two intersecting lines are. Spread is a number that goes from zero to one. A spread of zero means the lines are parallel and a spread of one means the two lines are perfectly perpendicular. Convert spread with our classical trigonometry, it is equivalent to square of the sine of an angle. So, S is equal to sine squared theta. So, spread of 1/4 represents what we classically call a 30° angle. Since sine of 30° is equal to 1/2, the square of that is 1/4. Here's how to calculate spread. If we have a right triangle, the spread between a leg and a hypotenuse is just the ratio of the quadrants. This leads us to the new and a bit cursed laws of trigonometry. Since the fundamental definitions of measurements have changed, the laws of trigonometry have to be rewritten. The result is a set of equations that only use basic algebra, addition, subtraction, multiplication, and division. No sines or cosines involved. From our familiar Pythagorean theorem, in this new realm, it becomes simple as there are no longer square roots. For a right triangle with legs Q1 and Q2 and hypotenuse Q3, it is written like this: Q1 + Q2 = Q3. As for what is equivalent to the law of sines, it becomes the spread law. The ratio of a spread to the opposite quadrants is constant for all three vertices of any triangle. As for what is equivalent to the law of cosines, we have the cross law. It looks like this. Remember, all of these do not require a calculator to find the cosine or sine of an angle. It only uses exact fractions.
So, in all, rational trigonometry hasn't caught on and replaced traditional classical trigonometry in high school. Largely because angles and distances are a bit more intuitive to the human brain. So, after all of this, are these trig functions really evil? Maybe they seem cursed to us because they've been hidden away from our high school textbooks. But, as we've seen today, many of these functions were simply born in different eras, designed to solve unique problems of their time. If you enjoyed going down this mathematical rabbit hole, hit the like button and make sure you subscribe so you don't miss the next video. Which of these cursed functions was your favorite? Let me know down in the comments. Thanks for watching, and I'll see you in the next one.