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.
.