More Functions/Trig :( 1.3/4

02/03/2026

how can you tell if two functions are inverses of each other?

• if they just have reversed x and y values of each other

• if they are reflected around the y=x line

how can you tell the inverse of a function will/won’t be a function

if the original function doesn’t pass the HLT, it’s inverse won’t be a function. (the HLT on the og func is the same as the VLT on the new func)

how can we write inverse functions?

f(x) » f^-1(x)

or in the case of trig functions, rather than sin^-1, cos^-1, and tan^-1,we can write arcsin, arccos, and arctan

note inverse tig funcs are not reciprocal trig funcs (not cosec, sec, and cotan)

review: tan is equal to

Y/X, or sin/cos

this also means sin = y, and cos = x

wat do we do to trig functions to ensure tat their inverses are functions?

we limit their domains, which in turn limits the ranges of their inverses

what domain limit do we put on cos waves?

typically, we do [0, π], which limits the range of arccos to [0, π]

(we use this limit to restrict quadrants later, also remember CAST rule)

what is the domain of ARCcos, and ARCsin

the max and min of cos’ range become the max and min of arccos’ domain, so its [-1, 1]

arcsin is the exact same process, as both waves have a max and min at the ones

what domain limit do we put on sin waves?

typically, we do [-π/2, π/2], which limits the range of arcsin to [-π/2, π/2]

(we use this limit to restrict quadrants later, also remember CAST rule)

if you ever see just an inverse trig func and a ratio, what are you looking for?

the angular measure. you should see arcsin(1/2) and think y = arcsin(1/2) and solve for y (which is an angular measure in radians, remember cast rule and limiting quadrants based on the limits of the inverse function)

note, this is often just a reference angle, so you may have to mess around and translate it into other quadrants

the hell do you do if you see this: sin(arccos(1/4))

1. ignore the outer sin, and then draw the inner ratio, (in this case (arccos(1/4)))

2. eliminate quadrants of the the graph based on cast rule, and then the limits of the graph, (in this case [0, π])

3. find the needed side, and then look back at the outer func, and write its ratio, (in this case sin = o/h = y/r = 15^1/2/4)