Trigonometric functions
The basics
Before we get to the basics of trigonometric functions, let's remind ourselves just what trigonometry is supposed to be.
Trigonometry is the study of triangles- more specifically, their angles and sides. They're used wherever the shoe fits because it is relatively simple and people are lazy. Hooray for laziness!
With that aside, what are trigonometric functions? Trigonometric functions are functions used for calculating the sides of a triangle, particularly right triangle (the law of sine and law of cosine allows you to break out of only using them for right triangles). An angle serves as input, and a corresponding formula serves as the output.
Hold your horses, I've heard of them- why are they called functions when they don't look like f(x)=y?
Remember how you can replace any of the letters used in f(x)=y notation with anything else? Well, the trigonometric functions, those being sine, cosine, tangent, and others, don't break that rule. The f in the notation is replaced with the name of trigonometric function, x with whatever angle you're dealing with, and y is the formula used to achieve your value. Take a look at the following formula (function): sin(35°) = (opposite/hypotenuse). When you replace sin with f, 35° with x, and (opposite/hypotenuse) with y, what does the formula (function) turn into?
Well, I see the unit circle being used in association with trigonometry. What's a circle doing in a triangle world?
Firstly, let's get a clear picture of what the unit circle is. The unit circle is a circle drawn on a coordinate plane with a radius of 1 (corresponding to the unit size), meaning whichever point on the circle line you choose to draw a line from to the origin point (0,0), you'll always get a radius of 1.
It is used for calculating values pertaining to angles, which, surprisingly, appear in any corner of a triangle. Once you understand what the unit circle is and what the associated terms are, it's simple and straightforward, which is why it's used in the realm of triangles.
Why do I keep seeing the names of trig. functions in problems that have nothing to do with angles and triangles?
That's because they're useful outside those contexts- areas that deal with periodicity, waves, etc. are the prized clientelle of trig. functions. In the context of schoolwork, a lot of exercises won't have you input actual numbers into the functions, but you'll be expected to be able to use formulas to transform and handle the abstract data provided.
Okay, I finally get it. So... what are the basic trigonometric functions again?
- Sine
- - Abbreviated as sin, it moves along the y axis. The right angled triangle definition of it is opposite over hypotenuse
- Cosine
- - Abbreviated as cos, it moves along the x axis. The right angled triangle definition of it is adjacent over hypotenuse
- Tangent
- - Abbreviated as tan or tg, it is the result of sine at a given angle divided by cosine at the given angle (sin/cos). The right angled triangle definition of it is opposite over adjacent
Other trigonometric functions
What is cotangent, secant, and cosecant supposed to be?
These are called multiplicative inverse or reciprocal (further reciprocals) ratios (denoted as 1/x or x-1 ).
Backtracking to the concept of reciprocals: reciprocals are numbers which when multiplied by its original number return 1. You obtain them either by raising the number to the power of -1 or by flipping it upside down when expressed in a ratio (this works for expressions as well). Take for example the number 3: its reciprocal would be 1/3 , because any plain integer can be said to be the dividend of the number 1. As for the ratio 4/8, the reciprocal would be 8/4.
This property can be used in equations and in times when you want to get rid of some really ugly numbers.
- Cosecant
- - Abbreviated as csc or cosec, it is the result of 1 divided by sine of x (1/sin(x)) (likewise, the reciprocal of sine is 1/cosec(x)). The right angled triangle definition of it is hypotenuse over opposite
- Secant
- - Abbreviated as sec, it is the result of 1 divided by cosine of x (1/cos(x)) (the reciprocal of cosine being 1/sec(x)). The right angled triangle definition of it is hypotenuse over adjacent
- Cotangent
- - Abbreviated as ctg, it is the result of 1 divided by tangent of x (1/tan(x)) (and the reciprocal of tangent being 1/ctg(x)). The right angled triangle definition of it is adjacent over opposite
What about arcsin, arccos, and arctan?
These are inverse trigonometric functions: sin-1(x), cos-1(x), and tan-1(x) respectively. They invert the functions. While you would use an angle in a, let's say, sine function, to derive a value, you'd use that value in an inverse function to derive the angle. For example: sin(90°) = 1; arcsin(1) = 90°
Inverses of the secant, cosecant, and cotangent exist, too.
Angles VS radians
What the hell is a radian and why would one use that instead of degrees?
Firstly, lets remind ourselves what pi is.
Pi is a ratio of a circle's circumference to its diameter (and the result is always the same, regardless of the radius or diameter of your circle)... that being approximately 3.14 (diameters). It is frequently written as the symbol π for the sake of convenience- that way, you don't have to write 3.14(..) every time you use pi, while also not rounding down its thousands upon thousands of decimal places (in theory).
Secondly, let's learn something about the ° symbol. People are mostly taught that it just means degrees and are left at that, however the degree symbol represents π/180 - that is, how many radians a single degree is equivalent to (this will make sense shortly).
With that out of the way, let's learn about radians. 1 radian is the central angle required to make an arc on a circle which's length is equal to the radius.
When looking at the unit circle in radians, you'll notice a bunch of values involving π in increments. Those are angles expressed in radians- 2π radians (~6.28 radians) being the same as 360°, π radians (~3.14 rad) the same as 180°, π/2 (~1.57 rad) same as 90°, so on and so forth. Once you get used to them, you should theoretically be able to use them as a quick way to get out of thinking when fit (author's note: I am not yet used to them, so I'm more likely to do things the hard way).
As for when would one use them- it depends on the circumstance. If you're simply doing math exercises or doing an exam, you have to look at the description of the problem or at the expressions themselves and check whether it mentions anything about using degrees or radians (or if the expression uses the degree symbol) and work from there. Additionally, calculus gravitates towards radians because a lot of rules only work for radians, so you'd most likely use radians there. When working with trigonometric functions, you'll most likely encounter degrees. In other circumstances, you'd probably know enough on the topic to be able to make the call.
Notes
- A lot of information in this entry was obtained from the video linked in the "further studying" section. Please have a look if you're stumped on the topic, it might be a bit lengthy for some but it's engaging and the explanations are easy to understand.
- 2 common symbols that are used to denote angles are α (alpha), β (beta), and θ (theta)
- Author's note: there wasn't any point during which secants and cosecants were brought up in my formal education, unlike cotangents... That, as far as I'm concerned, is the case for a lot of people from other countries, too. Supposedly, secants and cosecants are relics of the past- I don't know if that's true, but I can see niche use cases for them, so it might be good to be familiar with the concept simply for the sake of brevity.
- There's a whole bunch of stuff that may have been left out in the very brief explanation of arc functions. If necessary, the explanation will be revised in the future.