Is There a Trick to Remembering the Trigonometric Functions?

[jpcyiu]

hello everyone! I want to know how to verify cos sin tan
I always feel confused when i am doing the physics exercises.
are we always use cos when it is x-axis and use sin when it is y-axis??
I feel so confused.

 

[jedishrfu]

You just need to draw it out and keep playing with it until you know it inside and out.

Here's some info on polar coordinates where these often come up:

https://en.wikipedia.org/wiki/Polar_coordinate_system

 

[tony873004]

It's almost always cos for x and sin for y.
Snell's Law is an exception. You measure your angles off the normal rather than the x-axis. In either case, its pure soh cah toh. Don't memorize it. Like jedishrfu says, "just draw it out".

 

[SteamKing]

It's almost always cos for x and sin for y.

Except when it isn't. That's why you should memorize the definitions of sine, cosine, and tangent using a typical right triangle:

 

[rcgldr]

The old way "sohcahtoa" == sin => opposite over hypotenuse, cos = adjacent over hypotenuse, tan = opposite over adjacent.

 

[davenn]

haven't heard that one before

 

[sophiecentaur]

hello everyone! I want to know how to verify cos sin tan
I always feel confused when i am doing the physics exercises.
are we always use cos when it is x-axis and use sin when it is y-axis??
I feel so confused.

Since I was at school, I have been using a good 'cheat' method to check whether I should be using sin or cos. It avoids the 'x-axis / y-axis' question. What you do is to imagine that the angle in question is nearly zero (just imagine lowering the slope or altering the direction of the string etc.). Then decide whether the effect of the force would be to be small and increasing, as you increase the angle (in which case you choose sine) OR whether the effect of the force will be big and decreasing as you increase the angle (in which case, choose cosine). Try it for a simple example where you already know the answer and you will see what I mean. I think that advice helped a number of my students, who were confused in the same way.
Of course, there's no real substitute for drawing the proper vector diagram and really sticking to the rules you've been taught but the above method should give you confidence. There is also the problem of getting the right angled triangle the right way round (with the 90° in the right place). Just practice, is all I can tell you.

 

[Mark44]

The old way "sohcahtoa" == sin => opposite over hypotenuse, cos = adjacent over hypotenuse, tan = opposite over adjacent.

haven't heard that one before

It's been around as long as I can remember.

 

[sophiecentaur]

MR Worthington taught us: Some Old Hulks Carry A Huge Tub Of Ale. Could I ever forget that?

 

[sophiecentaur]

MR Worthington taught us: Some Old Hulks Carry A Huge Tub Of Ale. Could I ever forget that?
[Edit: Sorry about the repeat. My domestic wifi plays up sometimes.]

 

[Windadct]

I have very poor memory, I remember concepts / connection... so what worked for me was Remembering Sin (0) = 0 and Tan(0)=0

So Knowing Sin 0 = 0 means it must include the Opp in the Numerator ( as the angle goes to 0 - the Opp does, and you know nothing about the Adj and Hyp) - same for Tan (0) = Sin /Cos = again Opposite must be in the numerator, and thus Sin must be in the Numerator.. I know this is weird but there is no other solution. For example if Tan (0) = Cos/ Sin - and Sin(0) ... this would be undefined, etc...

This basic fact helped me test into Calc 1 - not having taken any Trig in HS.

 

[tfr000]

We learned "Oscar Had A Heap Of Apples".
But then you have to remember that the order is sin, cos, tan.

To understand and calculate trigonometric functions like cosine (cos), sine (sin), and tangent (tan), you need to have a basic understanding of right triangles and the relationships between the sides and angles. Here's how you can know and calculate these trigonometric functions:

1. Cosine (cos):
   • In a right triangle, the cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.
   • The formula for cos is: cos(θ) = adjacent side / hypotenuse.
   • To find the cos value of an angle, you need to know the lengths of the adjacent and hypotenuse sides of the triangle.
2. Sine (sin):
   • In a right triangle, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.
   • The formula for sin is: sin(θ) = opposite side / hypotenuse.
   • To find the sin value of an angle, you need to know the lengths of the opposite and hypotenuse sides of the triangle.
3. Tangent (tan):
   • In a right triangle, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.
   • The formula for tan is: tan(θ) = opposite side / adjacent side.
   • To find the tan value of an angle, you need to know the lengths of the opposite and adjacent sides of the triangle.

You can also use a scientific calculator or a computer software program to calculate these trigonometric functions for a given angle without the need to work with triangles directly. Just input the angle in degrees or radians, and the calculator or software will provide you with the corresponding cos, sin, and tan values.

Additionally, trigonometric functions have specific values for common angles (e.g., 0°, 30°, 45°, 60°, 90°, 180°) that you can memorize or look up in trigonometric tables for quick reference.

 

[davenn]

It's been around as long as I can remember.

Hi Mark

an American thing maybe ?

Some Old Hulks Carry A Huge Tub Of Ale.

like that one too with this topic and other basic stuff I have seen on PF over the years
I find it a bit bewildering thinking back and realising all these things I was never taught at school

Dave

 

[Mark44]

In reference to "soh cah toa"...

Hi Mark
an American thing maybe ?

It might be, but I don't know.

 

[aikismos]

SOH CAH TOA is definitely an American thing...

(http://www.slideshare.net/shmurray/sohcahtoa-indian-story)

SOHCAHTOA is a Native American.

 

[houlahound]

Some Old Hags Can't Hide There Old Age.That's how my math teacher taught it, schools were less than PC then.

 

[syhprum1]

At school I was taught "Peter has been here playing billiards"

 

[NascentOxygen]

Some Old Hags Can't Hide There Old Age.That's how my math teacher taught it, schools were less than PC then.

You'll ALWAYS remember that ditty?

 

[NascentOxygen]

At school I was taught "Peter has been here playing billiards"

How does that help?

 

[epenguin]

How do any of them help? Mystified.

 

[Mark44]

How do any of them help? Mystified.

Really?
They are mnemonic devices
SOH -- sin = opp/hyp
CAH -- cos = adj/hyp
TOA -- tan = opp/adj

Some Old Hags Can't Hide There Their Old Age.

Some Old Hags - same as above
Can't Hide - cosine, with 'c' for cosine and 'a' for adjacent, Hags for hypotenuse
Their Old Age - same as above

 

[Mark44]

At school I was taught "Peter has been here playing billiards"

How does that help?

I have no idea how that ties into sine, cosine, and tangent.

 

[NascentOxygen]

Really?
They are mnemonic devices
SOH -- sin = opp/hyp
CAH -- cos = adj/hyp
TOA -- tan = opp/adj

Some Old Hags - same as above
Can't Hide - cosine, with 'c' for cosine and 'a' for adjacent, Hags for hypotenuse
Their Old Age - same as above

My hint about the missing 'A' word was maybe too subtle.

Some Old Hags Can't Always Hide Their Old Age

 

[Mark44]

My hint about the missing 'A' word was maybe too subtle.

Yep, too subtle for me.

 

[Mark Harder]

SOH CAH TOA is definitely an American thing...

(http://www.slideshare.net/shmurray/sohcahtoa-indian-story)

SOHCAHTOA is a Native American.

Yeah, that's how I first heard it "Chief Soh Cah Toa". I guess somebody thought it sounded native American. Maybe there was such a person. But I I ignore the whole Indian thing. Just plain old soh cah toa works for me. As for x and y, forget that description. It's really a special case and it's misleading. Trigonometric functions relate to angles in a triangle or circular rotations of a vector about a point, and so forth. They're co-ordinate free, and that includes circular coordinates.

 

[Perq]

I don't know for the others, but I learned how to not mistake them the "practical way".

Inclined plane was the best way for me to understand how this stuff works.
All you need to remember is that
sin(0)=0
sin(90deg)=1
cos(0)=1
cos(90deg)=0
And then certain inclinations of the plane. (for example, cos(0)=1, would mean that mg=mgcos(0))

Hope it helps. :P (it did for me)

Also, forget a,b,c and so on, it changes from case to case, and it will only get more confusing as you go, so simply understand how this thing work, and not which letter corresponds to which.

EDIT: I just read that:

Since I was at school, I have been using a good 'cheat' method to check whether I should be using sin or cos. It avoids the 'x-axis / y-axis' question. What you do is to imagine that the angle in question is nearly zero (just imagine lowering the slope or altering the direction of the string etc.). Then decide whether the effect of the force would be to be small and increasing, as you increase the angle (in which case you choose sine) OR whether the effect of the force will be big and decreasing as you increase the angle (in which case, choose cosine). Try it for a simple example where you already know the answer and you will see what I mean. I think that advice helped a number of my students, who were confused in the same way.
Of course, there's no real substitute for drawing the proper vector diagram and really sticking to the rules you've been taught but the above method should give you confidence. There is also the problem of getting the right angled triangle the right way round (with the 90° in the right place). Just practice, is all I can tell you.

That is actually the same way I did it. :D There must be some universal truth to how this is a good way to learn it.

 

[aikismos]

I have no idea how that ties into sine, cosine, and tangent.

It doesn't. It ties to how mnemonic devices work. (See https://en.wikipedia.org/wiki/Mnemonic)

Mathematical inferences have to be drawn from mathematical facts. With incomplete facts, no valid reasoning no matter how strong the inferential engine.

Device: VerGraNuSym
Fact: The NCTM in the 1980's and 90's advocated that mathematics should be taught through the deliberative process of self-reflection in terms of Verbal, Graphical, Numerical, and Symbolic abilities instead of rote memorization of fact.

Device: SOH CAH TOA
Fact: There exist three fundamental functions called sine, cosine, and tangent (yes, which correspond to Euclidean constructs) which are defined in terms of the ratio of the sides of triangles.

Inference: To excel on standardized mathematical tests such as the ACT and SAT minimize the use of your graphing calculator without a CAS because the thrust of the assessment is to see if you can reason from the analytical geometrical perspective to the symbolic one equating facts such as how the ratio of the sine and cosine function are by definition the tangential function.

Bad student: Spends 2 minutes punching the lengths of the sides of the triangle and finally realizes that the items have no numbers in them before getting discouraged and moving on to the next problem to repeat the process.

Good student: Sets aside the calculator and sees that the 30-60-90 triangle has three distinct ratios (called SOH CAH TOA) which interrelate allowing her to see which of the four items is the only reasonable relation among them in less than 20s.

Not everyone has eidetic memory, so call it a mathematical strategy who doesn't enjoy the benefits of mild autism.

 

[Mark44]

At school I was taught "Peter has been here playing billiards"

I have no idea how that ties into sine, cosine, and tangent.

It doesn't. It ties to how mnemonic devices work. (See https://en.wikipedia.org/wiki/Mnemonic)

The question isn't "what is a mnemonic device?" Instead, the question is "what does the sentence about Peter playing billiards have to do with anything?" The thread here is concerned with three of the trig functions. To aid in understanding these ratios, several mnemonic devices have been posted.What is the point of what you wrote below? As far as I can tell, it is pretty much off-topic.

Mathematical inferences have to be drawn from mathematical facts. With incomplete facts, no valid reasoning no matter how strong the inferential engine.

Device: VerGraNuSym
Fact: The NCTM in the 1980's and 90's advocated that mathematics should be taught through the deliberative process of self-reflection in terms of Verbal, Graphical, Numerical, and Symbolic abilities instead of rote memorization of fact.

Device: SOH CAH TOA
Fact: There exist three fundamental functions called sine, cosine, and tangent (yes, which correspond to Euclidean constructs) which are defined in terms of the ratio of the sides of triangles.

Inference: To excel on standardized mathematical tests such as the ACT and SAT minimize the use of your graphing calculator without a CAS because the thrust of the assessment is to see if you can reason from the analytical geometrical perspective to the symbolic one equating facts such as how the ratio of the sine and cosine function are by definition the tangential function.

Bad student: Spends 2 minutes punching the lengths of the sides of the triangle and finally realizes that the items have no numbers in them before getting discouraged and moving on to the next problem to repeat the process.

Good student: Sets aside the calculator and sees that the 30-60-90 triangle has three distinct ratios (called SOH CAH TOA) which interrelate allowing her to see which of the four items is the only reasonable relation among them in less than 20s.

Not everyone has eidetic memory, so call it a mathematical strategy who doesn't enjoy the benefits of mild autism.

 

[aikismos]

The question isn't "what is a mnemonic device?" Instead, the question is "what does the sentence about Peter playing billiards have to do with anything?" The thread here is concerned with three of the trig functions.

What is the point of what you wrote below? As far as I can tell, it is pretty much off-topic.

Ah, I see your perspective now. I drew a different inference, and I'm ashamed I have no evidence to support it, but I took from the post a certain koan-like subtext. The OP:

hello everyone! I want to know how to verify cos sin tan
I always feel confused when i am doing the physics exercises.
are we always use cos when it is x-axis and use sin when it is y-axis??
I feel so confused.

We contributed a wide range of ideas including repetition (a good strategy), context (noticing the role it plays in force diagrams, and mnemonic devices (of which I cited the story behind SOH CAH TOA and through my post you question present a real life example of the efficacy mnemonics borne out by common experience in education) of which several were proffered from our international community here. That's a certain generality to the whimsical nature of them, and I took the intention of the OP to intentionally extrapolate a piece of nonsense which fits the general pattern, but which underlies the very advantage of mnemonics over repetition or a more gestaltist approach: namely, that's it's literally the very nature of the disconnect between the trigonometric precepts and the whimsy of the phrase which makes it superior to repetition (boring as hell) and framing (which is highly idiosyncratic for learners). Thus the emphasis of the importance of randomness and whimsy comes through heightening the randomness and whimsy to the point of total disconnect. But maybe the poster of said statement is just plain confused, and I was projecting my favored argument to persuade the OP onto meaningless form. We should add to this forum the ability poll... that would be interesting.

 

[Mark44]

Ah, I see your perspective now. I drew a different inference, and I'm ashamed I have no evidence to support it, but I took from the post a certain koan-like subtext. The OP:
We contributed a wide range of ideas including repetition (a good strategy), context (noticing the role it plays in force diagrams, and mnemonic devices (of which I cited the story behind SOH CAH TOA

You are aware, I hope, that the etymology you posted about SOH CAH TOA was completely made up.

and through my post you question present a real life example of the efficacy mnemonics borne out by common experience in education) of which several were proffered from our international community here. That's a certain generality to the whimsical nature of them, and I took the intention of the OP to intentional extrapolate a piece of nonsense which fits the general pattern, but which underlies the very advantage of mnemonics over repetition or a more gestaltist approach: namely, that's it's literally the very nature of the disconnect between the trigonometric precepts and the whimsy of the phrase which makes it superior to repetition (boring as hell) and framing (which is highly idiosyncratic for learners). Thus the emphasis of the importance of randomness and whimsy comes through heightening the randomness and whimsy to the point of total disconnect. But may the poster of said statement is just plain confused, and I was projecting my favored argument to persuade the OP onto meaningless form. We should add to this forum the ability poll... that would be interesting.

We have adopted tags recently that are intended to do just that. B - beginner, I - intermediate, A - advanced. We haven't used them much here in the math sections, but they are used a fair amount in the physics sections.