math mnemonics
- Thread starter galactus
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To remember which trig functions are positive in each quadrant,
\(\displaystyle \;\;\)remember "All Students Takes Calculus".
Write A-S-T-C in Quadrants 1 to 4 (in that order).
In Quadrant 1, "A" means All functions are positive there.
In Quadrant 2, "S" means the Sine (and its reciprocal, cosecant) is positive.
In Quadrant 3, "T" means the Tangent (and its reciprocal, cotagent) is positive.
In Quadrant 4, "C" means the Cosine (and its reciprocal, secant) is positive.
And, of course, this diagram indicates where the positive values are.
\(\displaystyle \;\;\)(There's a huge plus-sign in the center!)
\(\displaystyle \;\;\)remember "All Students Takes Calculus".
Write A-S-T-C in Quadrants 1 to 4 (in that order).
Code:
|
S | A
|
------+------
|
T | C
|In Quadrant 1, "A" means All functions are positive there.
In Quadrant 2, "S" means the Sine (and its reciprocal, cosecant) is positive.
In Quadrant 3, "T" means the Tangent (and its reciprocal, cotagent) is positive.
In Quadrant 4, "C" means the Cosine (and its reciprocal, secant) is positive.
And, of course, this diagram indicates where the positive values are.
\(\displaystyle \;\;\)(There's a huge plus-sign in the center!)
I assume that this information is too late for most of you,
\(\displaystyle \;\;\)but see what you think of my approach.
Being introduced to all six trig functions is always intimidating.
There is a list of six new words and their corresponding ratios to memorize.
I tried to make this task as painless as possible.
I begin with a circle of radius \(\displaystyle r\) at the origin.
I sketch an acute angle \(\displaystyle \theta\) in standard position.
Its terminal side interests the circle at a unique point \(\displaystyle (x,y)\).
We make ratios (fractions) with these three quantities: \(\displaystyle x,\;y,\;r\)
\(\displaystyle \;\;\) and give them names. \(\displaystyle \;\)(Derek, Heather, . . . just kidding!)
First, we memorize three new words: sine, tangent, secant ... in that order.
Each is followed by a "co-function": cosine, cotangent, cosecant.
Then we abbreviate these six names: sin, cos, tan, cot, sec, csc.
Write these in the first column.
(Then I write their definitions in the second column and \(\displaystyle \,x\;y\;r\,\) in the third.)
\(\displaystyle \L\;\;\sin\,\theta\;\;\frac{y}{r}\;\;\;\not{x}\;y\;r\)
\(\displaystyle \L\;\;\cos\,\theta\;\;\frac{x}{r}\;\;\;x\;\not{y}\;r\)
\(\displaystyle \L\;\;\tan\,\theta\;\;\frac{y}{x}\;\;\;x\;y\;\not{r}\)
\(\displaystyle \L\;\;\cot\,\theta\;\;\frac{x}{y}\)
\(\displaystyle \L\;\;\sec\,\theta\;\;\frac{r}{x}\)
\(\displaystyle \L\;\;\csc\,\theta\;\;\frac{r}{y}\)
For the \(\displaystyle 1^{st}\) ratio, cross out the \(\displaystyle 1^{st}\) letter \(\displaystyle (x)\)
\(\displaystyle \;\;\)and make a fraction of the remaining letters: \(\displaystyle \L\frac{y}{r}\)
For the \(\displaystyle 2^{nd}\) ratio, cross out the \(\displaystyle 2^{nd}\) letter \(\displaystyle (y)\)
\(\displaystyle \;\;\) and make a fraction of the remaining letters: \(\displaystyle \L\frac{x}{r}\)
For the \(\displaystyle 3^{rd}\) ratio, cross out the \(\displaystyle 3^{rd}\) letter \(\displaystyle (r)\)
\(\displaystyle \;\;\)and make a fraction of the remaining letters
The remaining letters are \(\displaystyle x\) and \(\displaystyle y\), but the fraction is not \(\displaystyle \L\frac{x}{y}\)
Instead, it is \(\displaystyle \L\frac{y}{x}\) . . . We must remember that.
\(\displaystyle \;\;\)A reminder: in the first two rows, the \(\displaystyle y\) is "above" the \(\displaystyle x\).
Then I point out that the last three are reciprocals of the first three.
\(\displaystyle \;\;\)And they are related by "nested arrows".
\(\displaystyle \L\;\;\sin\,\theta\;\;\frac{y}{r}\;\;\leftarrow ----*\)
. . . . . . . . . . . . . . . . . . . . . . . . .\(\displaystyle |\)
\(\displaystyle \L\;\;\cos\,\theta\;\;\frac{x}{r}\;\;\leftarrow --*\;\;|\)
. . . . . . . . . . . . . . . . . . . . . \(\displaystyle |\;\;\;|\)
\(\displaystyle \L\;\:\tan\,\theta\;\:\frac{y}{x}\;\;\leftarrow *\;\,|\;\;\,|\)
. . . . . . . . . . . . . . . . . . \(\displaystyle |\;\:\:|\;\;\;|\)
\(\displaystyle \L\;\;\cot\,\theta\;\;\frac{x}{y}\;\;\leftarrow *\;\:|\;\;|\)
. . . . . . . . . . . . . . . . . . . . . \(\displaystyle |\;\;\;|\)
\(\displaystyle \L\;\;\sec\,\theta\;\;\frac{r}{x}\;\;\leftarrow --*\;\;|\)
. . . . . . . . . . . . . . . . . . . . . . . . .\(\displaystyle |\)
\(\displaystyle \L\;\;\csc\,\theta\;\;\frac{r}{y}\;\;\leftarrow ----*\)
I ask them to practice writing the entire list from memory.
And I assure them that they will gradually become more familiar with them
\(\displaystyle \;\;\) so that this brute-force memorization will become unnecessary.
.
\(\displaystyle \;\;\)but see what you think of my approach.
Being introduced to all six trig functions is always intimidating.
There is a list of six new words and their corresponding ratios to memorize.
I tried to make this task as painless as possible.
I begin with a circle of radius \(\displaystyle r\) at the origin.
I sketch an acute angle \(\displaystyle \theta\) in standard position.
Its terminal side interests the circle at a unique point \(\displaystyle (x,y)\).
We make ratios (fractions) with these three quantities: \(\displaystyle x,\;y,\;r\)
\(\displaystyle \;\;\) and give them names. \(\displaystyle \;\)(Derek, Heather, . . . just kidding!)
First, we memorize three new words: sine, tangent, secant ... in that order.
Each is followed by a "co-function": cosine, cotangent, cosecant.
Then we abbreviate these six names: sin, cos, tan, cot, sec, csc.
Write these in the first column.
(Then I write their definitions in the second column and \(\displaystyle \,x\;y\;r\,\) in the third.)
\(\displaystyle \L\;\;\sin\,\theta\;\;\frac{y}{r}\;\;\;\not{x}\;y\;r\)
\(\displaystyle \L\;\;\cos\,\theta\;\;\frac{x}{r}\;\;\;x\;\not{y}\;r\)
\(\displaystyle \L\;\;\tan\,\theta\;\;\frac{y}{x}\;\;\;x\;y\;\not{r}\)
\(\displaystyle \L\;\;\cot\,\theta\;\;\frac{x}{y}\)
\(\displaystyle \L\;\;\sec\,\theta\;\;\frac{r}{x}\)
\(\displaystyle \L\;\;\csc\,\theta\;\;\frac{r}{y}\)
For the \(\displaystyle 1^{st}\) ratio, cross out the \(\displaystyle 1^{st}\) letter \(\displaystyle (x)\)
\(\displaystyle \;\;\)and make a fraction of the remaining letters: \(\displaystyle \L\frac{y}{r}\)
For the \(\displaystyle 2^{nd}\) ratio, cross out the \(\displaystyle 2^{nd}\) letter \(\displaystyle (y)\)
\(\displaystyle \;\;\) and make a fraction of the remaining letters: \(\displaystyle \L\frac{x}{r}\)
For the \(\displaystyle 3^{rd}\) ratio, cross out the \(\displaystyle 3^{rd}\) letter \(\displaystyle (r)\)
\(\displaystyle \;\;\)and make a fraction of the remaining letters
The remaining letters are \(\displaystyle x\) and \(\displaystyle y\), but the fraction is not \(\displaystyle \L\frac{x}{y}\)
Instead, it is \(\displaystyle \L\frac{y}{x}\) . . . We must remember that.
\(\displaystyle \;\;\)A reminder: in the first two rows, the \(\displaystyle y\) is "above" the \(\displaystyle x\).
Then I point out that the last three are reciprocals of the first three.
\(\displaystyle \;\;\)And they are related by "nested arrows".
\(\displaystyle \L\;\;\sin\,\theta\;\;\frac{y}{r}\;\;\leftarrow ----*\)
. . . . . . . . . . . . . . . . . . . . . . . . .\(\displaystyle |\)
\(\displaystyle \L\;\;\cos\,\theta\;\;\frac{x}{r}\;\;\leftarrow --*\;\;|\)
. . . . . . . . . . . . . . . . . . . . . \(\displaystyle |\;\;\;|\)
\(\displaystyle \L\;\:\tan\,\theta\;\:\frac{y}{x}\;\;\leftarrow *\;\,|\;\;\,|\)
. . . . . . . . . . . . . . . . . . \(\displaystyle |\;\:\:|\;\;\;|\)
\(\displaystyle \L\;\;\cot\,\theta\;\;\frac{x}{y}\;\;\leftarrow *\;\:|\;\;|\)
. . . . . . . . . . . . . . . . . . . . . \(\displaystyle |\;\;\;|\)
\(\displaystyle \L\;\;\sec\,\theta\;\;\frac{r}{x}\;\;\leftarrow --*\;\;|\)
. . . . . . . . . . . . . . . . . . . . . . . . .\(\displaystyle |\)
\(\displaystyle \L\;\;\csc\,\theta\;\;\frac{r}{y}\;\;\leftarrow ----*\)
I ask them to practice writing the entire list from memory.
And I assure them that they will gradually become more familiar with them
\(\displaystyle \;\;\) so that this brute-force memorization will become unnecessary.
.
- Joined
- Feb 4, 2004
- Messages
- 16,582
There's the mnemonic for keeping the basic metric prefixes straight:
. . . . .King Henry doesn't [usually] drink chocolate milk
...which, of course, stands for:
. . . . .kilo-, hecto-, deka-, [unit], deci-, centi-, milli-
You don't usually "need" the hecto-, deka-, or deci- prefixes, but knowing where they go ensures that the other prefixes are sufficiently far apart, so the decimal point gets moved the correct number of places.
Note: Many instructors leave out the "usually", so the students forget where the "unit" (the -meter, -litre, or -gram) part goes.
Eliz.
. . . . .King Henry doesn't [usually] drink chocolate milk
...which, of course, stands for:
. . . . .kilo-, hecto-, deka-, [unit], deci-, centi-, milli-
You don't usually "need" the hecto-, deka-, or deci- prefixes, but knowing where they go ensures that the other prefixes are sufficiently far apart, so the decimal point gets moved the correct number of places.
Note: Many instructors leave out the "usually", so the students forget where the "unit" (the -meter, -litre, or -gram) part goes.
Eliz.
skeeter
Elite Member
- Joined
- Dec 15, 2005
- Messages
- 3,216
always had a hard time spelling SOHCAHTOA, so here is one I learned many years ago while serving in the USN ...
read down the left column, then down the right column
\(\displaystyle \L Susie = \frac{Oscar}{Has}\)
\(\displaystyle \L Can = \frac{A}{Hat}\)
\(\displaystyle \L Tell = \frac{On}{Always}\)
the first letter of each word corresponds to the trig ratios ...
\(\displaystyle \L Sin = \frac{Opposite}{Hypotenuse}\)
\(\displaystyle \L Cos = \frac{Adjacent}{Hypotenuse}\)
\(\displaystyle \L Tan = \frac{Opposite}{Adjacent}\)
change one word, and you'll never forget it.
:wink:
read down the left column, then down the right column
\(\displaystyle \L Susie = \frac{Oscar}{Has}\)
\(\displaystyle \L Can = \frac{A}{Hat}\)
\(\displaystyle \L Tell = \frac{On}{Always}\)
the first letter of each word corresponds to the trig ratios ...
\(\displaystyle \L Sin = \frac{Opposite}{Hypotenuse}\)
\(\displaystyle \L Cos = \frac{Adjacent}{Hypotenuse}\)
\(\displaystyle \L Tan = \frac{Opposite}{Adjacent}\)
change one word, and you'll never forget it.
:wink:
B
brendan3eb
Guest
As far as remembering trig functions goes. I think it is best that kids learn the x/r type ratios instead of the opp/adj type ratios. I still don't know opp/adj, adj/hyp, etc. I just picture the triangle in my head, and think about which value is x and y. This may be easier for me though, because working with macromedia flash is one of my hobbies and I just think that way.
ultraviolet voodoo
I heard an odd one. It's odd enough that it's quite memorable, but unfortunately I already knew the equation when I heard the mnemonic. "ultraviolet voodoo" is the mnemonic for the integration by parts equation:
\(\displaystyle \int{u dv} = uv - \int{vdu}\)
ya' know, u v v d u. It makes sense to me anyway.
I heard an odd one. It's odd enough that it's quite memorable, but unfortunately I already knew the equation when I heard the mnemonic. "ultraviolet voodoo" is the mnemonic for the integration by parts equation:
\(\displaystyle \int{u dv} = uv - \int{vdu}\)
ya' know, u v v d u. It makes sense to me anyway.
For factoring the sum and difference of cubes, we have:
. . \(\displaystyle a^3\,+\,b^3\;=\;(a\,+\,b)(a^2\,-\,ab\,+\,b^2)\)
. . \(\displaystyle a^3\,-\,b^3\;=\;(a\,-\,b)(a^2\,+\,ab\,+\,b^2)\)
How do we remember the pattern of the signs? . Remember the word SOAP.
. . \(\displaystyle (a^3\,\pm\,b^3) \;= \;(a\,\pm\,b)(a^2\,\mp\,ab\,+\,b^2)\)
. . . . . . . . . . . . . . . . . .\(\displaystyle \uparrow\) . . . . . . .\(\displaystyle \uparrow\) . . . . \(\displaystyle \uparrow\)
. . . . . . . . . . . . . . . Same . . Opposite .Always Positive
Edit: Since the upgrade at this site, I have to realign my columns.
.
. . \(\displaystyle a^3\,+\,b^3\;=\;(a\,+\,b)(a^2\,-\,ab\,+\,b^2)\)
. . \(\displaystyle a^3\,-\,b^3\;=\;(a\,-\,b)(a^2\,+\,ab\,+\,b^2)\)
How do we remember the pattern of the signs? . Remember the word SOAP.
. . \(\displaystyle (a^3\,\pm\,b^3) \;= \;(a\,\pm\,b)(a^2\,\mp\,ab\,+\,b^2)\)
. . . . . . . . . . . . . . . . . .\(\displaystyle \uparrow\) . . . . . . .\(\displaystyle \uparrow\) . . . . \(\displaystyle \uparrow\)
. . . . . . . . . . . . . . . Same . . Opposite .Always Positive
Edit: Since the upgrade at this site, I have to realign my columns.
.
mammothrob
Junior Member
- Joined
- Nov 12, 2005
- Messages
- 91
not exactly a nemonic more of a rhyme, but for taking the derivative of a rational.
lo d hi minus hi d lo, square the bottom and off we go.
lo d hi minus hi d lo, square the bottom and off we go.
For the nine planets in order from the Sun:
. . Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Pluto.
. . \(\displaystyle \L\text{M}y\text{ V}ery\text{ E}arnest\text{ M}other\text{ J}ust\text{ S}erved\text{ U}s\text{ N}ine\text{ P}izzas\)
. . Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Pluto.
. . \(\displaystyle \L\text{M}y\text{ V}ery\text{ E}arnest\text{ M}other\text{ J}ust\text{ S}erved\text{ U}s\text{ N}ine\text{ P}izzas\)
D
Deleted member 4993
Guest
I used that stuff to remember history. Soon I needed "short-cuts" to remember my short-cuts.......
New mnemonic for planets, since Pluto's demotion: My Very Elegant Mother Just Served Us Nothing
My fave mnemonic for the chain rule dx/dy f(g(x)) = f'(g(x)) g'(x): douter(inner) dinner
Fun one for pi (count the letters in each word!): How I Want A Drink. Carbonated, Of Course.
Funny stupid mnemonic for the resistor color code: big brown rabbits often yield great big vocal groans when gingerly slapped = black, brown, red orange, yellow. green, blue, violet, gray, white, gold silver. (the initail letters of each phrase match up)
My fave mnemonic for the chain rule dx/dy f(g(x)) = f'(g(x)) g'(x): douter(inner) dinner
Fun one for pi (count the letters in each word!): How I Want A Drink. Carbonated, Of Course.
Funny stupid mnemonic for the resistor color code: big brown rabbits often yield great big vocal groans when gingerly slapped = black, brown, red orange, yellow. green, blue, violet, gray, white, gold silver. (the initail letters of each phrase match up)