# How did you remember sin, cos, tan ?

#### DarthVolta

Joined Jan 27, 2015
521
Was anyone one else taught

Some Old Hags
Can't Afford Husbands
Till Old Age

LOL

#### Glenn Holland

Joined Dec 26, 2014
705
Sin, Cos, and Tan very frequently pop up in physics and engineering so it's easy to remember their definition.

#### WBahn

Joined Mar 31, 2012
26,272
I don't really recall how I remembered them initially. I don't think it was with any silly saying (which have their place). But pretty quickly they were just things I knew.

I just recently saw this SOH-CAH-TOA thing which is something that is being used by kids now (and maybe for the last hundred or so years for all I know).

The thing that I struggled with for some time was remembering the definitions of secant and cosecant because they didn't seem to be coherently defined. For a while I couldn't make sense out of what the co- prefix meant, since cotangent was the reciprocal of tangent but cosine wasn't the reciprocal of sine. Then it was pointed out that the co- means that it is the value for the complementary angle, so at least that made sense. But I could never find out why secant wasn't defined as what cosecant is (and vice versa) as that seems to make more sense. I've since learned that the names were given by various people in a pretty haphazard fashion, so expecting them to make coherent sense is simply asking too much. So I finally decided that I would simply have to remember them and that the easiest way was to remember that the co- (not) meaning the reciprocal was actually useful and so I know that 1/sin is 1/co(something) and since it isn't cosine or cotangent (which I never had any problem remember IS the reciprocal of tangent), then the only thing left is cosecant. Not very elegant, but it's worked for me for the better part of four decades now allowing me to quickly figure out what the secant and cosecant are without ever having to look it up (which has been helpful since the handful of times I've needed it no reference was handy).

#### MrAl

Joined Jun 17, 2014
7,849
Hi,

There are a few things maybe a lot of things to remember about Sin, Cos, Tan.
Another one is what are the quadrants for the inverse functions. Think about that for a minute or so.
There are also a host of identities that are really good to know and trig substitutions for integrations.
It would be cool to start a list.

#### paulktreg

Joined Jun 2, 2008
790
It was SOH-CAH-TOA for me and that's 45 years ago.

#### ci139

Joined Jul 11, 2016
1,696
actually i didn't , but (some things i recall are) . . . Sine is "Odd" (symmetrical respective to Center of the Coordinates) while Cos is "Paired" (Symmetrical respective to Y axes) , Sin φ = y/R , Cos φ = x/R , ch x = (exp(x)+exp(-x))/2 , ch ix = cos x , sh x = (exp(x)-exp(-x))/2 , sh ix = i·sin x , sin x = (exp(ix)-exp(-ix))/(2i) , cos x = (exp(ix)+exp(-ix))/2 . . . sin α = cos (α - π/2) ← thus → sin (ß+π/2) = cos ß ← e.g. Sine is +90° phase-shifted Cosine . . . .

#### profbuxton

Joined Feb 21, 2014
419
How do I remember the trig ratios?
SOH CAH TOA. Acronym for "Sally opens her c@#% and her thighs open also.
works for me!

• BobaMosfet

#### joeyd999

Joined Jun 6, 2011
4,435
How do I remember the trig ratios?
SOH CAH TOA. Acronym for "Sally opens her c@#% and her thighs open also.
works for me!
Ah...not so progressive as I presumed.

#### SamR

Joined Mar 19, 2019
2,766
Some Old Hen and Catch A Hen. Tan is the other way. Op/Hyp, Adj/Hyp, so Tan is Op/Adj. I sent My Dear Aunt Sally to do it.

#### MrAl

Joined Jun 17, 2014
7,849
I think i have more trouble remembering all those acronyms than these:

sin: y/sqrt(x^2+y^2)
and:
cos: x/sqrt(x^2+y^2)
and
tan: y/x

or more simply from the circle with radius r at an angle A:
y=r*sin(A)
x=r*cos(A)
and
tan=sin(A)/cos(A)

which then can be written:
sin: y/r
cos: x/r
tan: y/x

• ci139

#### djsfantasi

Joined Apr 11, 2010
6,948
Ok, is there anyone from the dark side who also doesn’t have this problem? Either remember the definition or can derive it in a second.

#### OBW0549

Joined Mar 2, 2015
3,566
Ok, is there anyone from the dark side who also doesn’t have this problem? Either remember the definition or can derive it in a second.
I learned them by their definitions when I was taught them, back in 7th or 8th grade. It took me about 5 minutes to commit them to memory without any cute mnemonic tricks.

• djsfantasi

#### SamR

Joined Mar 19, 2019
2,766
You had far far better schools than I to be taught that in elementary school. All I had was very basic Algebra on the line of X + 2 = 5, what does X = ? I did get some basic number theory on different number bases.

#### MrAl

Joined Jun 17, 2014
7,849
Hi again,

As far as i remember i did not learn any tricks. When i first learned i started from the sine wave which has equation:
y=r*sin(wt)

and so that leaves the cosine wave:
x=r*cos(wt)

which is just x and cos instead of y and sin.
Tangent is just the division
(r*sin(wt))/(r*cos(wt))

so that's how i remember it.

Also of interest is that 'r' in those formulas is not always a constant but for many simpler problems it is so we would write:
y=R*sin(angle)
or
x=R*cos(angle)

I dont know if this is interesting to anyone but:
(x^2-y^2)/(x^2+y^2)=cos(2*angle)

perhaps for limited values.

#### WBahn

Joined Mar 31, 2012
26,272
Ok, is there anyone from the dark side who also doesn’t have this problem? Either remember the definition or can derive it in a second.
Me. As I mentioned earlier I just remembered which was sin(θ) on the unit circle and everything else pretty much falls from that. It was remembering which was sec() and which was csc() that took me some time.

Mnemonics have their place, particularly when there's a need to memorize arbitrary lists of things. But they are so frequently abused as a means of memorizing things that can be readily understood.

• djsfantasi

#### ci139

Joined Jul 11, 2016
1,696

#### Duane P Wetick

Joined Apr 23, 2009
432
From Popular Science...long ago...Big Chief SOH CAH TOA Remember 3,4,5 triangle gives angles of 36.9 & 53.1 degrees. Solve any trig. problem.

Cheers, DPW [Everything has limitations...and I hate limitations.]

#### MrAl

Joined Jun 17, 2014
7,849
From Popular Science...long ago...Big Chief SOH CAH TOA Remember 3,4,5 triangle gives angles of 36.9 & 53.1 degrees. Solve any trig. problem.

Cheers, DPW [Everything has limitations...and I hate limitations.]
Yes and also interesting somewhat is that 5 is an integer obtained from the two sides of the triangle: 5=sqrt(3^2+4^2).

#### WBahn

Joined Mar 31, 2012
26,272
Yes and also interesting somewhat is that 5 is an integer obtained from the two sides of the triangle: 5=sqrt(3^2+4^2).
What's interesting about it? That Pythagorean triples exist at all? Or that 3,4,5 happens to be one of them? If the latter, then consider that if it wasn't a Pythagorean triple, we wouldn't use it -- just like no one talks about 5,6,7 triangles but they do talk about 5,12,13 triangles.

That Pythagorean triples exist is not that interesting, either.

Pythagorean theorem: a² + b² = c²

Given any two integers m and n (with m>n), a = m² - n²; b = 2mn; and c = m² + n² is a Pythagorean triple.

• ci139

#### MrAl

Joined Jun 17, 2014
7,849
What's interesting about it? That Pythagorean triples exist at all? Or that 3,4,5 happens to be one of them? If the latter, then consider that if it wasn't a Pythagorean triple, we wouldn't use it -- just like no one talks about 5,6,7 triangles but they do talk about 5,12,13 triangles.

That Pythagorean triples exist is not that interesting, either.

Pythagorean theorem: a² + b² = c²

Given any two integers m and n (with m>n), a = m² - n²; b = 2mn; and c = m² + n² is a Pythagorean triple.

What is interesting about it is that it is rare among integers to get a result that is also an integer.
Here are four examples.
Using integers 1 through 10 we get 2 percent that work out to an integer.
Using integers 1 through 100 we get 0.63 percent that work out to an integer.
Using integers 1 through 1000 we get 0.1034 percent that work out to an integer.
Using integers 1 through 10000 we get 0.014474 percent that work out to an integer.
So it looks like as the number of integers we allow under the radical increases the lower the percent that work out to an integer after the square root is taken.
This is assuming that 3 and 4 would be considered the same as 4 and 3 for example otherwise double each result above.
It could be that this behavior levels off though i did not go any higher yet.