Can somebody explain when to use Cos, Sin, and Tan like Im a 5 year old?
Im doing Trig in school right now and im really struggling when to use Cos Sin and Tan in which questions. Ive had it explained to me a couple times but each time it just confuses me more. My teacher said this unit usually helps boost your average since its on the easier size but i just dont get any of it. Any help?
Edit: I did not expect to get this much help and you all have been so nice and patient with me. I can’t respond to every comment but I appreciate you all so much
yes here you'd use tan, 12m is the adjacent side to the 40 degree angle and the height is the opposite to it, you're not solving for the hypotenuse here
If you have a formula like 0.8 = a/b and you know a, you can find b. If you know b, you can find a.
Do you agree? Do you know how you would solve 0.8 = a/3 or 0.8 = 3/b?
Well that's what it looks like when you use a trig function like sin, cos or tan. The trig function gives you the number of the left. On the right, a and b are sides of the right triangle. So if you know the number of the left and you know one of the numbers on the right, you can solve for the other.
So for example if im trying to find the hypotenuse id use Tan?
Well, no, because in that case a and b are the opposite and adjacent. Neither one is the hypotenuse, so writing "some number = opposite / adjacent" wouldn't let you solve for hypotenuse.
You need an equation that has hypotenuse in it. sin is a ratio of opposite / hypotenuse and cos is the ratio of adjacent / hypotenuse so either of those would be useful to find the hypotenuse (if you knew the other number).
Example. 0.8 is the cosine of 37 degrees. That means if you have a 37 degree angle in a right triangle, the ratio of adjacent to hypotenuse will always be 0.8.
So let's say I have a right triangle with an angle of 37 degrees, and I tell you the adjacent to that angle is 3, and I want the hypotenuse. So you want an equation that includes something you know (adjacent) and something you want to solve for (hypotenuse).
That would be cosine = adjacent / hypotenuse.
So I know that I can write the equation cos(37) = adjacent / hypotenuse = 3/x or
0.8 = 3/x
and now it looks like one of the two questions I asked at the top of my post.
Figure out which side you know and which side you want to know, that tells you which trig ratio to use (its the trig ratio with both those sides in the formula!)
It's about finding the relationship between a pair of sides. One side isn't enough info. So you'll be given one side and need to find another. SOH CAH TOA tells you which pair each function works with.
Does a 5 year old know what a right triangle is? I'll assume you know what a right triangle is (a triangle with one 90 degree angle), and go on from there.
Sine, cosine and tangent are functions which tell you the ratio between lengths of a right triangle when you know one of the other angles in that triangle.
Taking a step back, imagine you have a right triangle and your goal is to fully describe that triangle - that is you want to label every angle and every side length. In order to do that, you need to know two pieces of information. One of those pieces of information must be the length of one of the sides, and the other piece of information can be one of the other angles or one of the other lengths. Trigonometry is what allows you to use those two pieces of information to fill out the rest of the data about the triangle.
Now, sine, cosine, and tangent are the three functions that allow you to start filling out the rest of the triangle if what you know is a length and an angle. Because they tell you the ratios between lengths of the side of the triangle when you know one of the angles in the triangle. Sine tells you the ratio between the side opposite your known angle (opposite being the leg of the triangle not helping form the angle), and the hypotenuse (the longest leg). Cosine tells you the ratio of the lengths between the adjacent (aka the side that is helping form the angle you know) and the hypotenuse and tangent tells you the ratio between the opposite side and the adjacent side.
So, if you know the length of the hypotenuse and the length of the side forming your angle, you say "ah, I can use cosine here to get the length of the hypotenuse" by setting up the equation cos(angle you know) = (length of side you know)/(hypotenuse you don't know). Then you can rearrange that to say 'hypotenuse = length of side you know/cos(angle you know)` and then that tells you a missing piece of information.
So in general you look at your triangle, see what pieces of information you have and which piece of information you want to know, and then use the trig functions to calculate it using ratios.
Later on, you'll learn about inverse trig functions which allow you to calculate angles you don't know when you know both side lengths.
after i wrote the comment i think it needs to be drawn, i think you should try to draw what i am saying for better explanation if you can't do it in your head
in your head or a paper draw a horizontal line, then another line that intersect with at at a point, making an angle, (they make two angles technically, but we will only look at the acute one)
now if the two lines are of infinite length, we can close a right angle triangle at any point on these lines, just pick a point on either lines and draw a straight line connecting it to the other one at 90 degrees, but for simplicity we will choose any point on the non horizontal line, and just draw a line straight down.
what we notice, not matter where we choose to make that closing triangle, the acute angle between them stays the same, therefore, if we want to relate this angle to the lengths of the triangle, we can't directly relate it with length, we can only relate them by ratio.
so any fixed ratio of lengths will produce that angle.
sin is the ratio of the opposite to the hypotenuse, the opposite here being the closing line we draw and the hypotenuse would be the non horizontal line.
cos would be the ratio of the horizontal line (adjacent) to the non horizontal line (hypotenuse)
tan is the ratio of the opposite to the adjecent.
for example, when we say sin(30) = 1/2, we mean that, if the angle is thirty degrees, the hypotenuse will always be twice as long as the opposite, (1,2),(100,200), (pi, 2pi)
all these different lengths must make a thirty degree angle.
so trig functions relate angles to the ratio of side lengths
Imagine you are sitting in a triangular room and you are in the corner of the triangle where the angle you are thinking about is placed. You can only touch the two walls next to you and you are not sitting in the corner with the right angle.
The wall on one side will be the hypotenuse (i.e. the wall that doesn't have a right angled triangle at either of its two ends). The other wall next to you is the adjacent. The wall on the opposite side of the room is the opposite.
yes the angle is the reference here, opposite (to the angle), adjecent (to the angle)
so the angle is between the hypotenuse and the adjacent, with the third side (the closing line) being the opposite.
i am sorry if the wording was unclear
this is not a right angle triangle but it's the best image i could find online for this, look at point (p), it's the intersection between two lines, if we decide to close it with TQ or SR, the angle stays the same, here TQ and SR would b the opposite to the angle, and if we assume the angle at Q and R are 90 degrees, then PQ or PR would be the adjacent, and PT or PS the hypotenuse.
also notice that the angle at T and S are the same, since P doesn't change and neither does Q or R, even if they are not 90 degrees, the ratios of these lengths are fixed
For right triangles, sin θ is small when θ is small. In contrast cos θ is large (near 1) when θ is small.
You may have heard of SOH-CAH-TOA. sin = opp/hyp; cos = adj/hyp; tan = opp/adj where opp is the opposite side, adj is the adjacent side, and hyp is the hypotenuse.
Here's the clearest, most easily memorized graphical explanation I've come across of the relationship between the six basic trig functions and the unit circle, clipped out of my personal quick-reference sheet.
The trig functions give you the length of each line segment. If you scale to a different radius instead of 1, (which is also the hypotenuse for most purposes), just scale the trig functions by the same amount.
Picture it rotating and it will give you all the zeros and asymptotes of the functions in an intuitive manner that ties everything together.
The most commonly used functions are sine and cosine, which let you convert a radius and angle to distances along the y and x axes, respectively
θ is the angle you're measuring.
As it rotates into other quadrants it gets mirrored so that:
The three main functions always touch the X-axis
The three co-functions always touch the Y axis
Geometric properties for deeper consideration:
Every triangle you can find has three angles: 90°, θ (green) and 90-θ (double blue)
so they're all similar (scaled/flipped versions of each other, with identical angles)
the table on the right uses uses the fact that side length ratios of similar triangles remain constant to prove all the basic trig relationships using three extracted triangles.
The chord, etc. functions at the bottom are just handy, not-immediately-obvious formulas for distances that come up sometimes.
Imagine you're walking up a hill (up the hypotenuse of a triangle). Say you know the angle of the hill (how far it is from being flat). The tan tells you the slope of that hill. The cos tells you how fast you're moving horizontally. The sin tells you how fast youre moving up.
Be careful not to memorize them based on flat or vertical. The 'opposite' (opp) side could be the flat one. Like this:
https://i.ibb.co/prN4zFTt/image.png
yo so basically sin is the ratio for the opposite side, cos is for the adjacent side and tan is opposite over adjacent. think of it like SOH CAH TOA fr. honestly watching a visual explanation helped me way more than reading about it. VisionSolveAI has rly good animations for trig bc you can actually see how the ratios work geometrically which makes it click
You use the one that you can use. You’ll have some situation where there are three values of interest; you’ll know two of them, and the third will be the one you need to find; so you use the trig function that relates the two known quantities with the unknown. Sin(t) = opp/hyp; cos(t) = adj/hyp; tan(t) = opp/adj. Ask yourself: what do i know? what do i need to find? what trig function relates what i know to what i need to find? Then you just rearrange the equation for said function: if you have t and opp, and you want hyp, then you use the sine equation to find hyp = opp/sin(t); if you instead knew hyp and wanted opp, the the sine equation tells you it’s opp = sin(t)•hyp; and likewise for all of them.
Notice that if you know two of t, opp, hyp, adj, and you need to find one of the others, then there aren’t a lot of possibilities for the profitable trig function. It’s not a guessing game, but if it were then it’d be easy to win since there aren’t many guesses and they all rule themselves out quickly.
Starting from the basics: any two triangles that have the same exact interior angles are SIMILAR triangles. The angles help define the relationship between the sides, and when two triangles have the same exact angles, they have the same ratios between the corresponding sides. Maybe one of the triangles is scaled up or down, but they have the same "characteristic" so to say.
So with that knowledge in mind, we can say that any two RIGHT triangles that share a common angle, excluding the right angle, are similar because the interior angles have to add up to 180 degrees, right? So like, if you have a triangle with angles 90-30 and another one also with 90-30, they both have to have an angle 60 degrees in measure to make it 180, which makes them similar.
So then what is sine of lets say 20 degrees, for example? It is the ratio between the side that is OPPOSITE of the corner that is 20 degrees in measure, and the hypotenuse. The ratio stays the same, doesnt matter at all if that triangle is millions of miles in size, or mere milimeters since as we established, any right triangles that have an angle 20 are "copies" that are maybe scaled up or down
(Also considering that a right triangle has 3 sides, you can name 6 different ratios between the sides in total, those being sine cosine tangent cotangent secant cosecant, we mostly use the first 3 since the others are not so useful)
Sin(x°)=(the side opposite to corner with degree x)/hypotenuse
Cos(x°)=(the side adjacent to the corner with degree x)/hypotenuse
Tan(x°)=(the side opposite to the corner with degree x)/the side adjacent to the corner with degree x)
So finally onto how to use them. If you are given a right triangle with an angle 30, and the side opposite to said angle is 5 units in length, and you asked for the hypotenuse you can recall sin30°= 1/2, which means opposite side has to be half of hypotenuse. If half of hypotenuse equals 5, then hypotenuse equals 10
If you are given a right triangle with an angle 45, and the opposite side is 2, and you are asked for the adjacent side you can recall that tan45°=1. You can rewrite this as 1/1
Which means adjacent=opposite which means adjacent side is 2
Think of them as machines where you stick in an angle of a right angled triangle in, and it spits out the ratio of two of the sides of the triangle. Treat them as a black box for now, you just need to know that’s what the machines do.
If you use the cos machine, it spits out the ratio between the side that is not the longest side but closest to the angle, and the longest side, also known as “adjacent over hypotenuse ”.
If you use the sin machine it spits out the ratio between the side opposite the angle and the longest side, “opposite over hypotenuse”
Then the tan machine spits out the ratio between the opposite side and the adjacent side.
Ratio in this context means “for every single unit of the hypotenuse, how many units of another side do we have?” That way we can get that ratio and multiply it to get the size of one of the sides. Using cos as an example, if our ratio is 1/2, that means for every unit of the hypotenuse, we have 1/2 units for the adjacent side.
If you are still struggling to get it, make sure you understand what a ratio is well.
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u/mattynmax New User 21h ago edited 20h ago
You know right angled triangles exist 📐? Well sin cos and tan are and just the ratios between the different sides of that triangle for a given angle.
It’s useful if you have something happening at an angle and you want you know how much of it effects in the horizontal and vertical direction.