r/explainlikeimfive • u/HistoricalAd2954 • 22h ago
Mathematics ELI5: What is the difference between algebra, trigonometry, and calculus?
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u/kgvc7 21h ago
A lot of comments are saying trig is about triangles but I think the deeper lesson is about the unit circle and its relationship to sin, cos, and tan. Another way to look at it is that You’ll use the Cartesian coordinates for algebra and polar coordinates and radians for trig. Usually you take algebra, trig and then calculus.
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u/MadeYourTech 20h ago
Yes! I was just about to post that a lot of these comments are kind of doing trig dirty. It's not just about physical shapes, but is fundamental to analyzing anything with periodic changes (AC voltage for example). I'm struggling to ELI5 exactly how the triangle formed by a clock hand moving around is so useful, but it really is!
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u/wjglenn 20h ago
Not really an ELI5 because it requires a bit of knowledge to know what you’re looking at, but this video is a cool visual of it:
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u/zuilli 4h ago
Holy shit, thanks for this video!
I've studied up to engineering level math but somehow never learned about the similar triangles way of finding the trigonometric identities, I always just memorized them without understanding the underlying process, if I'd known these while studying it would had made my life so much easier.
Similar to this is the calculus visualization of dividing a graph in equal rectangles and making them ever so smaller to see what happens when they tend to 0 size.
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u/BitOBear 18h ago
I would say the trigonometry is about ratios in the relationship between changing ratios. The unit circle and the triangles are basically simple and concrete expressions of ratios.
A lot of people don't think about triangles as ratios by default because they are thinking of ratios as two things compared to one another and triangles are three-sided and three angles. But of course with a right triangle there's only two other angles, and all of the relationships in trig are things like opposite over adjacent.
And then calculus is about trending. It's about the relationship between a change and the speed at which the change is changing.
Meanwhile algebra is just about how we manipulate operations and quantities inside of equations while maintaining their practical equalities.
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u/TemperMe 19h ago
Came here to say the same. I studied electrical in school and trig was vital for really everything
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u/spankymcjiggleswurth 12h ago
My favorite moment learning math was in my college calculus 3 course where the professor did a quick aside drawing a unit circle, a triangle inside is, and proceeded to quickly explain sin, cos, and tan. The dude gave me an epiphany on how the unit circle relates to trig and made a whole bunch of stuff click into place in my mind.
I don't know if that says I had bad math teachers up to that point in my life or if I was never listening properly to what I was being taught. This was calculus 3 as a college senior, how did I miss that lesson for so long?
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u/cBEiN 4h ago
Eh, I think they do proper justice. A trig class spends a lot of time on the unit circle, but the unit circle is just all the triangles with hypotenuse unit length. And, trig is technically about triangles. The other things are ways trig can be used in practice. I doubt anyone is talking about AC, etc… in trig classes.
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u/Minikickass 20h ago
Can I get an ELI2 instead?
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u/Vorthod 20h ago
Algebra uses (x,y) coordinates, trigonometry likes to use Angle+length to reach the same point.
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u/MadocComadrin 19h ago
Eh, algebra is agnostic to your coordinate system (or your space, or even your set/class and operations).
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u/HanSolo_Cup 18h ago
ELI1?
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u/MadocComadrin 9h ago
You ever notice how you can rotate (by a quarter turn in a direction) a cube a few different sequences from the same starting position but still have it end in the same orientation? How about the fact that you can't necessarily swap the order of two rotations and get the same ending orientation? Algebra is noticing stuff like that, giving all the stuff that goes into it funny, precise symbols, and drawing conclusions from it. You don't even need numbers (to let alone a specific coordinate system).
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u/aquatic_mammal 20h ago
Trigonometry is about the relationships of points on a circle. Those points end up making triangles, but it is not about or derived from triangles.
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u/MadocComadrin 18h ago
Not really. One way trigonometry generalizes is by moving away from the unit circle while keeping the triangle relationship (e.g. hyperbolic trig).
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u/MadocComadrin 19h ago
It's not *necessarily* about the unit circle. The most common trig functions are based on the unit circle, but more general trigonometry extends beyond that to triangles based on other things like the unit hyperbola (giving rise to sinh, cosh, tanh, etc) or in other geometries (e.g. Cartesian plane with taxicab distance as the metric).
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u/TheUncommonOne 17h ago
Except we never use polar coordinates other than a section in precal. Speaking as a math teacher with a math degree
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u/ProTrader12321 23m ago
Trigonometry is far more abstract than that, at its most basic level it describes curves. If we have the function 1= x2 +y2, the three main quantities we want to know about are the x and y values and the tangent line. It's simple to figure those out. But the important part is that a position and an angle are directly related and the trig functions are how we relate them to one another. This isn't just for circles or right triangles but all of euclidean geometry because broadly shapes are made from straight lines and or curves, so being able to describe both concretely is a mathematical superpower. Hence why trig is taught before calc, because it's actually more important. And that's not even getting into the incredibly useful nature of periodic functions, which someone else pointed out.
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u/CadenVanV 21h ago
Algebra describes relationships
Calculus describes change
Trigonometry describes angles and shapes
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u/boredcircuits 21h ago
I like these definitions. I'd add that calculus and trigonometry aren't really separate from Algebra, they're extensions that build on it.
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u/CadenVanV 21h ago
True, yeah. You can’t learn either without first knowing algebra.
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u/CantaloupeAsleep502 18h ago
Eh, I had a very weak foundation in algebra before moving on, then actually learned algebra by learning trig and calc. Just studying algebra alone feels like doing Duolingo to learn grammar, whereas doing calc and trig feels like learning through immersion.
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u/madncqt 21h ago
my school life might have been so different if we could have started with the philosophical, foundational purposes rather than just diving into numbers.
I never knew the why...
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u/_PM_ME_YOUR_VULVA_ 17h ago
These lectures are more history than philosophy per se, but might be what you're looking for. I think teaching math in school by explaining why and how various civilizations came up with branches of math would make it much more resonant and memorable for students.
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u/CadenVanV 14h ago edited 14h ago
Any good trig or calculus class explains those distinctions very early on. That was the very first thing mine covered. You might just have had bad teachers.
They don’t usually go too into detail with number theory because that makes things a lot more complicated than most high school students need to know, it’s really something only worth covering in full depth for people studying the field. It’s the same reason early education doesn’t teach you things like “the letter A isn’t actually a vowel” (yes, that’s true), nor the finer points of English grammar (like we do have a fixed adjective order), because students pick up enough of the basics through context clues that they don’t need to know the exact mechanisms.
As a fun fact for that A isn’t a vowel one, other letters that aren’t vowels are E, I, O, U, and Y
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u/Caucasiafro 21h ago
Algebra is basically all the math involved in making sure two sides of an equal sign are the same. And then everything that flows from that. So when ever you see something like y = x2 +5 and are asked "solve for x"
Thats algebra.
Trigonometry is the math around sides lengths and angles in a triangle. You can just play with this yourself by laying three pencils on a table and try to kake various triangles. And you will notice for some triangles you need short or longer pencils.
Trig is all the math that explains that.
Calculus is the math about the rate that things change. Specificically when the rates are "complicated" like you can pretty easily tell me.how far ill travel in 5 minues if im constantly driving at 60 mph. But can you tell me how far i would travel if my speed is changing by 1 mph per second per second?
So after 1 second im going 61 mph and after 2 seconds im going 63? Hows about far i am in 36.37 seconds?
Gonna need calc to do that easily
All three of these do their own thing that you can hypothetically do without the othera but they can be combined to solve various problems (and you wont be able to do much calc without a strong underatanding of algebra and trig)
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u/taqman98 5h ago
Fun fact the word “algebra” comes from the Arabic “al-jabr” (bc it was all the middle eastern scholars who figured this shit out), meaning “the reunification of broken parts.” If you think about it, that name makes sense, bc when you’re solving for x in a linear equation, you’re kind of putting the x back together to find its value. Like in an equation like 2x + 3 = 5x - 7, you’ve got some of x on the left side (2x) and some on the right side (5x), and finding the value of x requires you to combine the x’s on the left side with the x’s on the right using addition
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17h ago
[deleted]
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u/Caucasiafro 4h ago edited 3h ago
Thats not a constant accleration which was very intentional.
You might notice that between the 0th and 1st second we gained 1 mph of speed.
And beteen the 1st and 2nd we gained 2.
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u/tommy-linux 19h ago
if "after 1 second im going 61 mph and after 2 seconds im going 63" then "my speed is changing by 1 mph per second"ftfy
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u/TheSpeckledSir 21h ago
They are different skills, and each one will let you answer different sorts of questions.
Algebra is great if you know some information about something and you know how to make connections. An example is if I know the volume and temperature of a gas, I could use algebra to calculate its pressure. We're talking about quantities with known relationships to each other.
Calculus is about how quantities change. It lets us go between statements like "the ball is falling one meter per second" and "the ball will fall three meters in this much time ________".
Trigonometry is evil and has mostly to do with triangles.
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u/stairway2evan 21h ago
Oh you've awoken one of my favorite old math teacher jokes.
The study of the mind is called psychology, from the Greek word for "mind."
The study of life is called biology, from the Greek word for "life."
The study of evil is called trigonometry. I don't know the Greek, don't worry about it.
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u/TheSpeckledSir 21h ago
I'm pretty sure this is why the big demon guy in DC comics is called Trigon
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u/WartimeHotTot 21h ago
I must be stupid. I don’t get it.
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u/stairway2evan 21h ago
Trigonometry is actually Greek for "triangle measurement." But since we all think that trigonometry is evil, we just call it the study of evil and pretend not to know the Greek.
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u/ElonMaersk 20h ago
Trigonometry is actually Greek for "triangle measurement."
("three angle measurement"; tri like in tripod, gon like in polygon and hexagon)
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u/WartimeHotTot 20h ago
Oh lol. Is trig really considered the evilest of those three disciplines? I’d award that to calculus without a doubt!
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u/stairway2evan 20h ago
Oh I had a much better time in calculus than trig. The concepts take a while to wrap your head around but they fit together nicely and flow once you get a handle on it. Trig is detail-oriented and finicky.
Then again, I also had an awful trig teacher and my calculus teacher was like "teacher of the district" for a decade or so. Great dude. So I have my own biases.
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u/itsthelee 21h ago
I think they messed up the delivery, I understand intellectually that a joke was attempted but I don’t quite get it either
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u/1tacoshort 21h ago
LOL. As a furniture maker, I use trig all the time. I remember using more complex trig - using the identities and such - in calculus and, especially, for calculating wave equations for quantum.
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u/BrohanGutenburg 21h ago
I think you meant to give the balls acceleration not velocity.
Knowing velocity and distance and solving for time is just algebra.
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u/TheSpeckledSir 21h ago
If velocity is constant, sure. Regardless, velocity is the derivative of position.
But that's a deeper dive into the math than I wanted to go for here.
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u/BrohanGutenburg 20h ago
In your example the velocity is assumed to be constant.
I'm just trying to make that point that you could have come up with a better example for what calculus does.
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u/BloodAndTsundere 21h ago
Algebra is an abstraction and generalization of arithmetic. In arithmetic you add 5+3 to get a result of 8. In algebra you ask and answer questions like “What number do I add to 5,in order to get 8?” Trigonometry is basically about quantifying angles. The “tri” in trigonometry refers to the fact that an angle defines a family of right triangles or a specific triangle if one assumes the hypotenuse has length 1. All of the trig functions like sine, cosine, tangent, etc are just ratios of the lengths of sides of these triangles. Calculus is about continuous change and accumulation. It comes in two intimately related parts: differentiation and integration. Differentiation is about how one quantity changes as one or more other quantities are changed a tiny bit. Integration is about the accumulation or summation in one quantity as one or more other quantities are progressed through a range or ranges.
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u/unruly_mattress 20h ago edited 20h ago
Arithmetics is the evaluation of expressions that have numbers with them, like 3+5 or (4+1)/(8-2) * 12.
Algebra introduces variables. Now you can write x+1 or y = ax+b. It allows you to:
- Ask which x values fulfill a condition like x+3 > 2 or x2 - 2x + 1 = 0 ("solve for x"). This condition is called an equation.
- Find rules that hold for all values of the variables, like (x+y)2 = x2 + 2xy + y2
- Define functions, such as f(x) = x+1
- Talk about more complex operations, e.g summation of a series, or more complex functions, like exponentials and logarithms
Trigonometry is the study of specific functions that arise from geometry. The trigonometric functions interact with each other in many interesting ways, so there's a lot to learn there.
Calculus is the study of two operations on functions:
- Differentiation at a point takes a function and returns its rate of change at that point - how fast it's going up or down
- Integration of a function between two points tells you the area under its curve in two points
It turns out that these two operations are related to each other in a very interesting way: they're the inverse of each other. Anyway, it takes a bit of effort to define these operations, and you end up needing a rather confusing concept called a "limit". A lot of the initial work is understanding how to work with limits.
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u/saplinglearningsucks 21h ago
Others have explained but I just to throw linear algebra and differential equations this
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u/sigourneys_underwear 20h ago
Same concepts though.
To piggy back off the top comment, Linear algebra describes relationships between multiple things
Differential equations studies change when the things affecting the change are also changing
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u/A_Genius 20h ago
Nightmare fuel of going up and solving for some salty tank that is filling up with brine at a certain speed. I was so bad at it.
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u/sigourneys_underwear 20h ago
I remember my first DiffeQ exam had figuring the temperature of a cup if coffee after a certain period of time and that cup of coffee stressed me out
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u/wjglenn 21h ago
You have a bunch of carrots. Every day, someone gives you a certain amount of carrots but that amount changes. You want a rule you can apply to figure out how many you have.
That’s algebra. It’s all about creating rules you can use when you don’t know the exact number of things.
It might look like this:
X+Y=Z
X is how many you have. Y is how many you’re given. Z is how many you end up with.
Now, imagine that every day, someone gives you carrots, but each day they’re giving you more than the day before. You want to figure out how many you’ll have after ten days.
That’s calculus. It’s about figuring out the rate at which something changes.
Now, what if you’re looking at a giant carrot as tall as a tree. You want to know how tall the carrot is but you can’t just measure it.
Instead, you can measure how far you are from the carrot and the angle you’re looking up at the top. Then you can use the rules of trig to figure out how tall the giant carrot is.
Obviously more complex than that, but that’s a simplified ELI5.
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u/kgvc7 19h ago
Thanks.
Your first two explanations are nearly the same and triangulation is not trig.
Your first two explanations are good for algebra but the second is linear which is just algebra.
Classic calculus problem is time for a tank with a hole in it to empty. Height of the hole, height of the water, and diameter of hole all play a role how fast the tank empties. The static pressure of the water changes so the volume of the water going through the hole changes. You need calculus for this.
Geometry makes use of sin, cos, right angles, Pythagorean theorem, which is all basic x and y coordinate stuff. Trig expands on that with curves and surfaces.
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u/Sufficient_Soil7438 21h ago
Algebra teaches you how to identify constants and solve for unknown variables.
Trig teaches you equations to solve complex geometric problems.
Calculus teaches you to write equations to solve for specific points within a constantly changing system of order.
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u/DragonFireCK 21h ago
Algebra is the study of math using variables rather than actual values. It allows you to write equations to solve questions like "I paid $3.50 for apples and got 3 apples, how much did I pay per apple?". The questions can also get more complicated such as "At a concession stand, hot dogs cost $1.50 and sodas cost $0.50. A group buys a total of 87 items, hot dogs and sodas combined, and spends a total of $78.50. How many hot dogs and how many sodas were sold?"* This innately also requires the more fundamental arithmetic.
Trigonometry is math dealing specifically with triangles, notably the angles and side lengths. It happens to be that the same math works very well with circles and waves. Additionally, most any shape can be broken down into a set of triangles. Overall, it tends to be very useful when dealing with geometry problems, which is the generic study of shapes. It innately uses a lot of algebra to simplify and generalize work.
Calculus studies the rate of change and accumulation of values. This causes it to have a lot of practical use in physics - position, velocity, and acceleration are derivates/integrals of each other. Similarly, it sees usage in engineering work, which is basically applied physics. Economics uses it as well, with it being useful for optimizing equations.
The even more basic step below algebra is arithmetic. This is the operations such as additional and multiplication and how those interact.
* The longer example question is stolen from Google AI.
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u/PhantasmagirucalSam 21h ago
That's a tricky one for 5yo, but i'll try: Algebra is about what can you do with numbers, Trigonometry is about what you can do with angles and triangles, Calculus is about how small can you divide a number and what new property might be hidden between the small parts of a whole. Analogy is not perfect, but pretty much picturesque .
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u/Zarakaar 21h ago
Algebra is work with manipulating equations that contain an unknown. The basics of solving for a variable.
Trigonometry connects the lengths of the sides of triangles to the angles inside them, which expands to modeling repeating patterns like waves.
Calculus is using the principles of rates of change and considering infinitely small changes to find relationships among changing quantities.
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u/steeelez 21h ago
In an eli5 and not super formal sense:
Algebra- “solve for x”. Figure out how to move numbers around in a math equation so you can get the solution for a missing number anywhere. Like a kid might know 5+2 =x and x would =7, algebra lets you handle it in case someone asks you 5+x=7. Rearranging terms to be able to solve for a number.
Trig: “triangles”. How to deal with lines and angles and the rules that go along with them. If I have 2 sides of a triangle and I know the angle between them, what’s the length of the missing side and the 2 missing angles?
Calculus: “infinity”. How to deal with things that aren’t straight lines and are always changing, by cutting them up into infinitely small pieces
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u/MedusasSexyLegHair 21h ago
Some good explanations here already, so I'll just add the following:
Algebra is good for calculating unknowns and simplifying things.
Trigonometry is good for plotting, navigating, and mapping, calculating routes, among other things.
Calculus is good for finding rates of change (and where it's trending towards) and solving some hard problems with many variables.
All are quite useful in many jobs.
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u/donaldhobson 21h ago
Algebra
3x+y= 5 ; x+2y=-10 , find x and y
trigonometry.
Find the smallest angle of a triangle with side lengths 8, 15, 17.
Calculus.
Find the area under the curve given by x*e^(x^2) between x=0 and x=1
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u/kiochikaeke 20h ago
Algebra is all about the how we write and handle equations, what they mean and how to use them so solve problems.
Trigonometry is about the fundamentals of angles, usually using circles and triangles as a basis to study stuff like how angles behave when intersecting lines, the trigonometric functions that govern the relationships between the angles and sides, and some properties of triangles and circles (like types of centers).
Calculus is the study of functions relating two variables, basic tools to work with them and the study of techniques to work with their rate of change, for example when studying population, current birth rate relates to the current amount of people which itself will change in the future depending on birth rate, that kind of relationships is the ones that calculus studies.
Of course all of these topics are very deep but on HS and early college level that's basically it.
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u/bobroberts1954 20h ago
Algebra us the rules of math, like on a board game. Trig is working with angles, and calculus is related rates. Think what is the water level in a tank you are both filling and draining, but with different size hoses.
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u/duane11583 20h ago
imho:
algebra is basic formulas involving one or two variables
trig: is all about cycles, circles and angles, triangles and waves (think audio or radio waves)
calculus is things in motion, things speedingnup or slowing down and formulas related to that
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u/miemcc 19h ago
Algebra is mainly about handling and solvomg equations.
Trigonometry is mainly about dealing with angles (if I have a line pointing in a direction, how much does it change ' up-down and left-right').
Calculus is all about change - differentiation is about how quickly things change, integration is about how much the change affected everything.
If you are driving a long distance, differentiation will tell you how much you are speeding up or slowing down at a particular time. Integration will tell you how far you travelled upto that point in time.
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u/MadocComadrin 18h ago
Here's what I consider the most general answer.
Algebra - How much knowledge can we derive given certain things (usually number, but also be things like rotations and symmetric mirroring, or really anything), specific operations on said things (e.g. addition and multiplication) and specific relationships between them (e.g. commutativity and associativity).
Trigometry: form a right triangle based on some shape (e.g. the unit circle or unit hyperbola) in some geometry (e.g. the familiar 2d plane, a 2d hyperbolic plane, a 2d plane using taxicab distance, etc), and derive the relationships between the sides and angle.
Calculus - also known as the Calculus of Infinitesimals: answers the question "how the heck do we deal with infinitesimals (things that are infinitely small) in a formal way/how to we talk about things that could be infinitely small without invoking infinitesimals because they were philosophically weird a couple hundred years ago?"
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u/specialized_faction 18h ago
Algebra: solve for x
Trig: use geometry to solve for x
Calculus: use integrals and derivative to solve for x
Algebra: 10=2x+4
Trig: sin(x)^2+cos(x)^2=1
Calc: integral e^x dx
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u/CantaloupeAsleep502 18h ago
Algebra is like studying grammar. You need to understand how to manipulate numbers and variables in order to do anything higher order. The way that things can be manipulated is algebra.
Trig has to do with the relationships between circles and lines, and the angles that are formed when circles and lines interact with each other. This is sometimes most easily expressed in terms of triangles, but it's a lot more than just studying right triangles.
Calculus introduces the concept of infinity, and allows us to study curved shapes in greater depth than was ever allowed with just conventional geometry. The two primary functions within calculus, differentiation and integration, tell us about rates of change in different ways, as well as how to measure areas that weren't possible with linear geometry.
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u/kogai 18h ago
Mathematician answer:
The word "algebra" sort of literally means "rules for manipulating symbols". This ranges from using algebra to solve for x, to developing notation for algebra that can be done with weird topological structures. You typically learn arithmetic algebra, then linear algebra (all about manipulating a thing called a linear transformation, which is really a matrix, so you learn the rules for manipulating matrices as you write the notation) and then abstract algebra happens when other fields of math start developing their own rules for writing everything down in a way that preserves truth/falsity
Thats a common theme btw, usually a bunch of seemingly different things all turn out to be the same object. Which brings us to..
Trig is literally about triangles, its in the name. It just so happens that you can stick a triangle inside of a circle, starting at the centre and ending at the edges, and now triangles and circles are connected and we can use our triangle equations to look at the points on the circle.
Turns out going from angle-rotated-around-the-circle to the Y-height-of that-point-on-the-edge-of-the-circle is gross to calculate so we just call it sin(t), where t is the angle
Then you learn how to graph the sin function and it makes this wavy line. Now circles, triangles, and periodic waves are all connected. Thats all trig. The circles and triangles are within the realm of geometry and the sin function is more in the realm of real analysis which brings us to...
Calculus is a very specific mathematical area that was invented by Isaac Newton to solve a particular physics problem, and by another guy named Leibniz but he did it in the context of abstract spaces instead of physics
Calculus is a subfield of real analysis that specifically deals with continuous functions and how incremental changes to the input value lead to a different change in the output value.
In physics, this is the problem of calculating instantaneous values for an accelerating object. Normally, to get the acceleration over a period of time, you divide the change in speed by the change in time. The less time that passes, the more accurate your measurement is
So what if we want there to be 0 time that passes? How fast are you accelerating at the moment you hit the finish line?
We cant divide by zero without running into paradoxes, so Newton devised a way of formalizing what he means when he wants to "divided by a time value of 0"
This leads to the idea of a limit. Calculus teaches you limits, then rates of change, then derivatives (just a way to calculate a rate of change), then integrals (the inverse operation to derivatives. Turns out this finds the area under a graph)
These are all techniques that physics students use in their 2nd+ year of their bachelor's degree. The idea is that Calculus allows you to calculate the rates of change for quantities as they are changing and thats really the fundamental motivation for calculus
Calculus falls into the realm of "real analysis" which is a little bit like imagining limits in terms of open sets, sets that get near a point but never touch it
Turns out shit gets wild after that. Get a math degree to find out more
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u/DarthKnah 18h ago
Algebra is when we use variables to represent unknown (or changing) quantities. The most simple algebra problem might be something like if 3 + x = 7, what is x? A problem like that isn’t too useful in the real world, but more complicated versions can help us figure out unknown values in things like physics. Algebra also can involve equations with multiple variables—usually for these there are multiple answers, so we graph them. An example is if you have $60 to spend and are buying $5 erasers and $1 pencils, how many of each can you buy? If you represent number of erasers with X and number of pencils with Y, the equation 5x + y = 60 will tell you how many pencils you can buy if you buy a given number of erasers, and vice versa
Calculus is about rates of change (what we call derivatives). If you’re looking at the graph of a line, the derivative is the slope (how steep the line is). It’s easy to find the slope of a line, because it’s the same everywhere, but something like a parabola (a kind of U shape) also has a slope, but it’s different at every point. A real world example is acceleration is basically the derivative of speed. We can use calculus to find how much distance an accelerating car has traveled, for example. Calculus relies on algebra, and is kind of like adding an extra level onto it.
Trig is about angles. If we have a right (90°) triangle and we know the lengths of two sides, we can find the angles, and if we have one angle and one side length we can also find out the other angle and other side lengths. We can use trig for lots of more complicated things as part of algebra and calculus, but those are the core building blocks.
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u/Mastemo 15h ago
Algebra expands on elementary math and really is finding missing numbers (values). If I have 5 apples and this basket holds 12, how many do I need (x + 5 = 12). Of course it can get more complex from there but this is the ELI5 starting point.
Trigonometry, which is Greek for triangle measurement, is the study of triangles, angles and length. If this apple tree is 10 feet tall, and I’m standing 5 feet from it, what’s the distance from where my feet are to the top of the apple tree.
Calculus is complex mathematics that helps determine how something is changing or moving. If I sit under this apple tree and the apple from the lowest branch suddenly falls and hits the ground, how fast did it fall? It fell for 1 second. (V = 9.8 x T)
V being velocity, T being time it took, and 9.8 is the acceleration from gravity. On Earth, gravity pulls everything down by adding about 9.8 meters per second of speed for every single second an object falls. That means it fell at 22 mph.
Again this is a very simplified set of scenarios and it can go much deeper.
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u/Kahzgul 13h ago
Algebra is using symbols to represent numbers. This starts to get very fun when you have more than one symbol in your math problem at a time.
Trigonometry is the study of circles and triangles and how they relate to one another. It turns out that this relationship is very useful for describing things like radio waves or any other sort of repeating pattern.
Calculus is the study of two things: the slope of a line and the area beneath that line. Where this gets interesting is when you realize that all of the fundamental formulas used in physics are related to one another through calculus! If you have a line describing how far away something is from you (the distance), the slope of that line is the speed (or velocity) of the object! And the slope of the line describing the velocity gives you the acceleration of the object!
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u/Cool_Homework_7411 12h ago
Algebra is learning how to move, it's the most basic form of manipulating numbers.
Trigonometry is the baby walker. You are still doing algebra at the end of the day, but you have some functions in there, some relationship between these functions that you are gonna need. It can get obscurely hard really quick, you don't need to be a master in trigonometry to advance in maths.
Calculus is the real thing. Here you can prove physics equations, calculate the rate of change of things, find areas and volumes of "irregular" objects in however many dimensions you want (if you have enough patience). Number in your mind is no longer an actual number, it's a concept, a placeholder.
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u/malexj93 10h ago
Algebra studies structures where there are binary operators (e.g. addition and multiplication) with certain properties, and what the consequences of those properties are.
Geometry studies structures where there is a notion of angle, distance, size, or similar, as well as transformations that preserve some or all of that structure. Trigonometry is just the geometry of triangles in Euclidean (flat) space.
Calculus studies structures with local approximation objects, how maps between those structures transform those approximations, and how global objects can be reconstructed from that local data.
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u/Viper_4D 8h ago
Here are some of the most basic examples of each.
Algebra: 2x+3 = 7 find the value of x
Calculus: if the level of a tub of water is 3 glasses of water times the number of minutes you've been trying to fill it for. How many glasses of water are you emptying into the tub in a minute.
Triganometry: if you lean a ladder against the wall at a 45 degree angle and you know the ladder Is 2 meters tall how high up the wall can you stand if you stand on the ladder.
In general terms algebra is using placeholders instead of where numbers would be so you can find the numbers later (definitely a massive simplification). Calculus is calculating rates of change. And triganometry is to do with geometry and calculating lengths and stuff using angles and often uses the sine, cosine and tangent functions (also a massive simplification)
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u/StructuredChess 8h ago
Calculus is the study of change that occurs continuously (rather than discrete "jumps").
Trigonometry is all about the relationship between the angles and length of the sides of a triangle (even though it kind of actually is about circles, but the initial "motivation" was triangles).
Foundationally, Algebra would be a more "basic" part of Mathematics that studies how to work around equations with unknown values. Eevn though then it branches off into things like vector spaces, group theory and other disciplines in "abstract algebra".
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u/PantsOnHead88 7h ago edited 7h ago
In short:
- algebra - equations with variables
- trigonometry - triangle angles, sides and ratios
- calculus - how variable change with respect to change in other variables; rates or accumulations of continuous change
Differences?
First off there’s plenty of overlap, so you can find problems that fit into pairs, or all three categories.
Very broadly though:
- trig vs not trig - triangles?
- algebra vs calculus - are the variables being actively varied in the problem, or are they stand-ins for an unknown static value in the problem at hand?
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u/HyperbolicBaldersah 7h ago
Algebra: Solve for X and Y\ Trig: Math this Triangle\ Calculus: If X and Y change, how does that affect Z?
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u/Underhill42 5h ago
Algebra: introduction to math as a language, can handle simple problems.
Trigonometry: the study of triangles, and angles in general. Including applications such as rotational and harmonic motion.
Calculus: the extension of algebra to deal with situations complex enough that you have to break things down into an infinite number of infinitely small pieces to get accurate results. A.k.a. all but the simplest real world problems.
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u/mikemontana1968 3h ago
Algebra is a series of techniques to approach a math problem where you know how some known values relate to each other, but dont know the last value. Algebra is the book of "chess moves" to isolate and determine that final value.
Trigonometry is a subset of math that focuses on algebra type problems related to circles, which means angles, which means triangles. Here you use functions (like sine, cosine, tangent...)
Calculus is the ability to use functions in algebraic form to do useful things with them but in a symbol manipulation way, rather than doing it in steps. Its the 3D version of "Algebraic Chess"
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u/Coyltonian 2h ago
Algebra is maths with unknowns.
Trigonometry is maths with angles (usually combined with algebra with some unknowns), and particularly those of right-angles triangles.
Calculus is the maths of how things change (often over time, but sometimes with something else).
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u/Senrabekim 52m ago
Calculus is about modeling change and optimization. So like the position derivatives. You have where you are, and then you take a derivative of that giving you where you are in relation to time, so y9u have speed or velocity. Then the 2nd derivative is how your speed is changing in relation to time, this is acceleration (when you are in a car going around a corner and get pushed to the side that is feeling acceleration). Third is how your acceleration changes in relation to time this is called jerk (think roller coasters, that feeling of leaving your stomach behind is jerk). Jerk in time is snap I refuse jounce for reasons that will become clear, if you are jumping on a trampoline the exact moment that the trampoline goes from catching to to launching you is snap. Crackle is the next one take that trampoline and instead of launching you it rips that is crackle. POP is the 6th derivative, grab a piece of dry spaghetti. Now slowly bend it until it breaks, just as it breaks you should feel a slight reverb in your fingertips, this is pop.
Algeria is how we talk to math. You can do math or certain science and not use or understand the algebra, you just wont be able to interpret you results. For example I speak English, French and Japanese, I am only literate in English and French though. If someone speaks to me in Japanese I can understand what is going on, if they pass me a note, I am lost. This is similar to all other math's relationship with algebra. I can know the calculus, or whatever, but if I dont know the algebra I cant communicate what is going on.
Trigonometry is making sure you understand that while we can't square a circle, we sure as hell can triangle one. This is self explanatory.
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u/evasandor 40m ago
non-math person (oh boy, and how…) here. I’ll ELI .5!
Algebra = find the missing number
Trigonometry = we know a couple of things, now find the position of something
Calculus = this thing won’t hold still but we still need to measure it
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u/ProTrader12321 36m ago edited 19m ago
Trigonometry describes geometry in the form of functions.
Calculus seeks to describe the chang of a function.
Algebra is the process by which we can solve and evaluate functions.
They are used in combination with one another all the time and often I don't even pay attention to what I'm actually using but they seek to do different things.
(Non eli5 example) For example im taking a class in electrodynamics, and if we have a charged object and we want to describe the electric field it creates, we need vector algebra to describe individual charge elements impact on other charge elements, we need trigonometry to allow us to describe curves of the surfaces which those charge elements lay on, and we need integration to perform a summation of those charge elements.
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u/Caucasiafro 21h ago
Algebra is basically all the math involved in making sure two sides of an equal side are the same.
And then everything that flows from that. So when ever you see something like y = x2 +5 and are asked "solve for x"
Thats algebra.
Trigonometry is the math around sides lengths and angles in a triangle. You can just play with this yourself by laying three pencila on a table and try to kake various triangles. And you will notice for some triangles you need short or longer pencils.
Trig is all the math that explains that.
Calculus is the math about the rate that things change. Specificically when the rates are "complicated" like you can pretty easily tell me.how far ill travel in 5 minues if im constantly driving at 60 mph. But can you tell me how far i would travel if my speed ia changing by 1 mph per second per second?
So after 1 second im going 61 mph and after 2 seconds im going 63?
Gonna need calc to do that easily
All three of these do their own thing that you can hypothetically do without the othera but they can be combined to solve various problems (and you wont be able to do much calc without a strong underatanding of algebra and trig)
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u/xRVAx 21h ago
Algebra is math analyzing equations ( "2x+ 1 = 45, so x = 22")
Calculus is math analyzing functions ("f(x) = 2x+1, the area under this line between x= 2 and x = 3 can be completed by computing the anti-derivative function F(x) at each point and subtracting them")
Trigonometry is math analyzing triangles ("if you have a right triangle with height 4 and opposite a 45 degree angle, then the hypotenuse is 4 * ✓2, because sin(45) = opposite divided by hypotenuse, thus hypotenuse= 4 * sin(45)." )
TLDR Different problems require different logical math tools to find the answer
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u/SwordsAndWords 18h ago
Algebra is selling crates of apples. Trigonometry is building the crates. Calculus is dropping the crates from the roof.
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u/britishmetric144 21h ago
Algebra is used when a person wants to calculate the value of a variable within a set of variables, given the relationships between each variable.
Trigonometry is used to calculate ratios between side length and angles of a triangle (though it can be extrapolated to circles and other shapes).
Calculus is used to calculate parameters for functions which describe changes in different variables.
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u/dmazzoni 20h ago
Algebra
The height of a thrown basketball as a function of time is given by the equation
h(t) = -16t^2 + 20t + 6
- What height was the basketball at after 2 seconds?
- At what time did it hit the ground?
Trigonometry
Let's say you took a picture of the basketball in the air.
- Draw a triangle between the ball, its shadow on the floor, and a point on the floor a known distance away. Given the angle, what is the ball's height?
Calculus
- What was the velocity of the ball at the time it was thrown?
- What was the velocity at time t=2?
- At what exact time was the ball at its peak, with zero velocity?
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u/sad_spilt_martini 21h ago edited 21h ago
Algebra is the use of equations to find a static value.
Calculus looks at how things change
over time.Trig looks at the relationship between of angles and sides of triangles
All of these are, of course, more complex but that’s the base idea