sin
Sine
aka "sinusy" · the y-coordinate girl
The Originals
sin(θ) = y = opposite/hypotenuse
okay so sin thinks she's SO normal and relatable. starts at zero, goes up to one, comes back down, goes negative, comes back. like girl we GET IT you oscillate. she's literally just tracking height as you walk around a circle but she acts like it's so deep. also she's ODD which honestly tracks — sin(−x) = −sin(x). very dramatic. very "i contain multitudes." she does have the best zeros though (0, π, 2π) very clean very aesthetic.
Period
2π
Range
[−1, 1]
Amplitude
1
Zeros at
nπ
Peaks at
π/2, 5π/2...
Personality
Odd function
cos
Cosine
aka "the x-coordinate" · sneaky little bitch
The Originals
cos(θ) = x = adjacent/hypotenuse
first of all she starts at ONE when sin starts at zero. like she couldn't even wait, had to be ahead immediately. she's x which doesn't even LOOK like x but apparently that's her whole identity. then she has this cute QI and QII restriction for her inverse like she's so special. and don't even get me started on sum/difference — cos(A+B) subtracts when you add and ADDS when you subtract. complete chaos. she's the EVEN function (cos(−x) = cos(x)) which means she's literally the same forwards and backwards. symmetrical. of course she is.
Period
2π
Range
[−1, 1]
Amplitude
1
Zeros at
π/2 + nπ
Starts at
cos(0) = 1
Personality
Even function 💅
tan
Tangent
aka "so/cute" · sin's little ratio · chaotic neutral
The Originals
tan(θ) = sin/cos = y/x
tan is literally just sin divided by cos. she has no original content. AND she has the audacity to be UNBOUNDED — no amplitude, no max, no min, just goes to infinity whenever cos hits zero. like ma'am you cannot just go to infinity that is not a personality. her period is π which is HALF of sin and cos because she completes her whole dramatic arc twice as fast. she has ASYMPTOTES which are basically just walls she throws herself against repeatedly. also odd. of course. and she equals 1 at π/4 which is the one moment she's actually chill.
Period
π (not 2π!!)
Range
(−∞, ∞)
Amplitude
none. NONE.
Asymptotes
π/2 + nπ
Zeros at
nπ
tan(π/4)
= 1. her one win.
✂️ the reciprocals (lesser known, equally messy)
csc
Cosecant
aka "1/sin" · sin's weird shadow
The Reciprocals
csc(θ) = 1/sin(θ) = hypotenuse/opposite
csc is literally just sin's shadow. everywhere sin is zero, csc has an ASYMPTOTE because you can't do 1/0. she never gets to be between −1 and 1 — she's always |csc| ≥ 1, which means she's more extra than sin in every direction. she looks like sin's graph flipped inside out. asymptotes where sin has zeros. it's giving villain origin story.
= 1/sin
that's it
Asymptotes
at nπ (sin's zeros)
Range
(−∞,−1]∪[1,∞)
Period
2π
sec
Secant
aka "1/cos" · cos's drama twin
The Reciprocals
sec(θ) = 1/cos(θ) = hypotenuse/adjacent
sec is cos but make it unhinged. asymptotes where cos has zeros (π/2 + nπ), so basically right where tan also blows up. sec and tan are in cahoots. sec starts at 1 (because cos starts at 1 and 1/1 = 1) and immediately starts doing the most. also even function because cos is even and sec is just cos in a trench coat.
= 1/cos
that's it
Asymptotes
π/2 + nπ
Range
(−∞,−1]∪[1,∞)
Personality
Even (like cos)
cot
Cotangent
aka "1/tan" aka "cos/sin" · tan's nemesis
The Reciprocals
cot(θ) = cos/sin = 1/tan
cot is tan but she does everything opposite out of spite. tan goes up from left to right — cot goes DOWN. tan has asymptotes at π/2 + nπ — cot has asymptotes at nπ (sin's zeros). tan has zeros at nπ — cot has zeros at π/2 + nπ. they share a period of π which is the only thing they agree on. cot is basically tan's evil mirror twin and honestly she's lowkey more organized because her asymptotes are at cleaner values.
= cos/sin
tan flipped
Asymptotes
nπ (tan's zeros)
Zeros at
π/2 + nπ
Slope
decreasing (tan is petty)
🚫 the inverses (restricted for a reason)
sin⁻¹
Arcsin
aka sin⁻¹ · "what angle gave me this"
The Inverses
arcsin(x) → angle in [−π/2, π/2]
arcsin had to be RESTRICTED because sin kept giving the same output for multiple angles and arcsin couldn't handle the commitment issues. so now arcsin only accepts angles from −π/2 to π/2 (QI and QIV only). she takes a y-value and gives you back the angle. very responsible. very "i only date people in my range." her output is always between −π/2 and π/2, no exceptions, even if you come at her with a cute 5π/4.
Input (domain)
[−1, 1]
Output (range)
[−π/2, π/2]
arcsin(1/2)
π/6
arcsin(−1)
−π/2
cos⁻¹
Arccos
aka cos⁻¹ · "QI and QII only, thank you"
The Inverses
arccos(x) → angle in [0, π]
arccos is cos but she only outputs [0, π] — QI and QII. not QIV like arcsin. she and arcsin are NOT the same even though people constantly mix them up which is honestly offensive to both of them. arccos(0) = π/2. arccos(1) = 0. arccos(−1) = π. she takes an x-value and gives back the angle. she's the most normal inverse honestly. her restriction is clean. good for her.
Input (domain)
[−1, 1]
Output (range)
[0, π]
arccos(1/2)
π/3
arccos(0)
π/2
tan⁻¹
Arctan
aka tan⁻¹ · "open range, closed personality"
The Inverses
arctan(x) → angle in (−π/2, π/2)
arctan takes ANY real number as input (tan's range is all reals so arctan's domain is all reals — she has range trauma) but her OUTPUT is strictly (−π/2, π/2). notice the parentheses — open interval, she never actually reaches the endpoints because tan's asymptotes live there. she approaches π/2 forever and never gets there. it's giving unrequited. arctan(1) = π/4. arctan(√3) = π/3. arctan(−1) = −π/4. she's the most useful one on exams because she shows up in triangle problems constantly.
Input (domain)
(−∞, ∞)
Output (range)
(−π/2, π/2)
arctan(1)
π/4
arctan(√3)
π/3
📋 sum & difference formulas (the gossip)
sin — plays it straight
sin(A + B) = sinA cosB + cosA sinB
sin(A − B) = sinA cosB − cosA sinB
sign in the middle matches the sign in the angle. boring. predictable. good.
cos — does the opposite just to be different
cos(A + B) = cosA cosB − sinA sinB
cos(A − B) = cosA cosB + sinA sinB
SUBTRACTS when you add angles. ADDS when you subtract. sneaky little bitch behavior confirmed.
how to remember sin(105°) on the exam
105° = 60° + 45°
sin(105°) = sin60 cos45 + cos60 sin45
= (√3/2)(√2/2) + (1/2)(√2/2) = (√6 + √2)/4
split into angles you know → plug in → multiply fractions → combine over common denominator