Did you still remember Special Right triangles? 45-45-90 and 30-60-90? If not, just remember this.
| 45 | 45 | 90 |
|---|---|---|
| leg | leg | leg√2 |
| 30 | 60 | 90 |
|---|---|---|
| leg | leg√3 | 2leg |
Did you found this familiar? Just to remind you, it is just a review.
This is a mnemonic about the three main trigonometric ratio. (sin, cosine and tangent
$$ sin(\theta)=\frac {opposite} {hypotenuse} $$
$$ cos(\theta)=\frac {adjacent} {hypotenuse} $$
$$ tan(\theta)=\frac {opposite}{hypotenuse} $$
Attention!!! This is new. There are three additional trigonometric ratios (cosecant, secant and cotangent), these are defined as the reciprocals of the original ratios (sine, cosine and tangent)
$$ csc(\theta)=\frac 1 {sin(\theta)}=\frac {hypotenuse} {opposite} $$
$$ sec (\theta)=\frac 1 {cos(\theta)}=\frac {hypotenuse} {adjacent} $$
$$ cot(\theta)= \frac 1 {tan(\theta)}=\frac {adjacent} {opposite} $$
Attention!!! Reciprocals are not the inverse function of an graph. The inverse of $f(x)=x+1$ is $f^{-1}(x)=x-1$, instead of $\frac 1 {x+1}.$ Just like exponent and logarithmic function, $f(x)=2^x$↔$f^{-1}(x)=log_2(x)$ not $y=\frac 1 {2^x}$.
https://www.desmos.com/calculator/uthingnlwr
| Original function | Inverse function(arc) | inverse function$f^{-1}(x)$ | reciprocals | inverse of reciprocal (arc) |
|---|---|---|---|---|
| sine(x) | arcsine | $sin^{-1} (x)$ | cosecant | arc-cosecant |
| cosine(x) | arcosine | $cos^{-1} (x)$ | secant | arcsecant |
| tangent(x) | arctangent | $tan^{-1} (x)$ | cotangent | arctangent |
Since y and x are flipped, just pug in the outcome to the inverse function, just like exponent and logarithmic function