Foundations — Inverse trig functions — arcsin, arccos, arctan — domain, range, graphs

This page assumes you have seen nothing. Before you can read the parent note, you need the vocabulary below. Each item gives you three things: what it means in plain words, what picture it corresponds to, and why the topic needs it. They are ordered so each one leans only on the ones above it.

1. Angle — and why we measure it in radians

Picture: stand at the centre of a circle, point straight right (this is angle $0$), then rotate anticlockwise. How far you've rotated is the angle.

We measure that turning in radians, not degrees. One radian is the angle you sweep when the arc you trace along a circle of radius $1$ has length $1$.

Key conversions you will see constantly (here $\pi$ always means $\pi$ radians, not the bare number — the little word "radians" is part of the value):

$180°=\pi \text{ radians},90°=\frac{\pi }{2}\text{ radians},60°=\frac{\pi }{3}\text{ radians},45°=\frac{\pi }{4}\text{ radians},30°=\frac{\pi }{6}\text{ radians}$

2. The unit circle and the two coordinates $\cos$, $\sin$

Now the two most important machines in the whole topic. Rotate by angle $\theta$ (the Greek letter "theta", our standard name for an unknown angle). You land on a point of the unit circle. That point has two coordinates:

• its horizontal position is called $\cos \theta$ ("cosine of theta"),
• its vertical position is called $\sin \theta$ ("sine of theta").

Figure s01 — sine and cosine are just coordinates. The violet ring is the unit circle. Turn by the angle $\theta$ (navy arc near the centre) and you land on the magenta dot. Read off two numbers: how far right you are (magenta segment, $\cos \theta$) and how far up you are (orange segment, $\sin \theta$). The new insight: $\sin$ and $\cos$ are not mysterious — they are literally the two coordinates of that dot.

Why the topic needs this: the parent note's whole story is "$\sin$ eats an angle and spits out a ratio." That ratio is literally a coordinate — a height between $-1$ and $+1$. This is why the domain of $\arcsin$ is $[-1,1]$: a height on the unit circle can never exceed $1$ or fall below $-1$. See it directly on the figure — the magenta dot's height is trapped between the top and bottom of the circle.

3. Tangent — the "steepness / slant number"

Picture: draw the line from the origin out to the point on the circle. $\tan \theta$ is the slope of that line — how steeply it rises. Flat line: $\tan =0$. Line at $45°$: rises as fast as it runs, so $\tan =1$. As the line tips towards straight-up, it gets infinitely steep.

Figure s02 — tangent is the steepness of the ray. Three rays leave the origin: the orange (gentle) has a small slope, the magenta sits at $45°$ so its slope is exactly $1$, and the navy (near-vertical) has a huge slope. The new insight: as the ray tips towards straight up, its slope races off to infinity — this is why $\tan$ has no ceiling.

4. Function, domain, range

Picture: a box with an in-slot and an out-slot. Domain = the pile of tickets you may drop in the in-slot. Range = the pile of tickets that ever come out.

Why the topic needs this: the parent note constantly states "Domain: $[-1,1]$, Range: $[-\pi /2,\pi /2]$." Those are just which tickets go in and which come out of the inverse machine.

5. One-to-one (invertibility) — the crucial obstacle

Picture: if you could draw a horizontal line that cuts the graph in two places, those two inputs give the same output — that breaks one-to-one. A one-to-one graph passes this "horizontal line test."

Figure s03 — the horizontal line test, and the safe piece of sine. The violet wave is $\sin x$ across many turns. Drop one orange dashed horizontal line at height $0.5$ and it stabs the wave in two navy dots — two different angles with the same sine, so sine is not one-to-one. The thick magenta stretch is the chosen safe piece $[-\pi /2,\pi /2]$ where the curve only climbs, so any horizontal line hits it just once. The new insight: restricting to that magenta stretch is exactly what rescues invertibility.

6. Monotonic (increasing / decreasing)

Picture: a hill you walk up left-to-right (increasing) or a slope you walk down (decreasing). A function that only ever climbs, or only ever descends, automatically passes the horizontal-line test — so it is one-to-one.

Why the topic needs this: the parent note picks each inverse's range by choosing a monotone piece: sine increasing on $[-\pi /2,\pi /2]$, cosine decreasing on $[0,\pi ]$, tangent increasing on $(-\pi /2,\pi /2)$. Monotone = safe = invertible.

7. Inverse notation: $\arcsin$, $f^{-1}$, and the $⟺$ symbol

The parent note writes statements like $y=\arcsin x⟺\sin y=x.$ Read the symbol $⟺$ as "means exactly the same thing as" (each side implies the other). So the line says: "$y$ is the arcsine of $x$" says precisely the same thing as "the sine of $y$ is $x$." One is the forwards machine, the other is backwards.

Now we can name the exact safe zones (domain = what goes in, range = the allowed answers) for all three inverses. Every value is in radians:

Notice each range is exactly the monotone stretch we chose in Section 6 — the safe zone is the range. The symbol $\mathbb{R}$ just means "every real number," matching the fact that a slope (Section 3) can be anything.

8. Reflection across the line $y=x$

Figure s04 — the inverse graph is a mirror image. The orange dashed diagonal is the line $y=x$. The magenta dot $(a,b)$ sits on the forwards machine $f$; flip it across the diagonal (navy dotted connector) and it lands on the violet dot $(b,a)$, which sits on the inverse. The new insight: running a machine backwards is swapping the two coordinates, and swapping coordinates is a mirror flip across $y=x$.

9. Odd / even functions and reference angles (assumed extras)

Picture: odd = rotate the graph $180°$ about the origin and it lands on itself; even = mirror across the vertical axis and it lands on itself. $\sin$ and $\tan$ are odd; $\cos$ is even. The parent note uses this to get $\arcsin (-x)=-\arcsin x$.

Picture: put the angle on the unit circle and drop a straight line from the point to the horizontal axis. The little wedge that forms at the axis is the reference angle. For example the angles $150°$ and $210°$ both lean the same amount off the horizontal, so they share the reference angle $30°$ ($=\pi /6$). The full-size angles differ only by which quadrant (which quarter of the plane) they live in, which fixes the signs of $\sin$, $\cos$, $\tan$.

Why the topic needs this: to compute something like $\arccos (-1/2)$ you first recognise the reference angle ($\pi /3$, since $\cos (\pi /3)=1/2$), then use the sign and the required range $[0,\pi ]$ to land in the correct quadrant ($2\pi /3$). Reference angles are the bridge from "I know the size" to "I know the exact angle." Build them fully in Unit circle and reference angles and lean on the Trigonometric identities.

Prerequisite map

Equipment checklist

Test yourself. Each line below is written as prompt ::: answer — the ::: is a reveal marker: read the part before it, answer aloud, then uncover the part after it to check. (It is just this vault's hide/show syntax, not any mathematical symbol.)

Connections

• Parent topic (Hinglish)
• Unit circle and reference angles
• Trigonometric identities
• One-to-one functions and invertibility
• Derivatives of inverse trig functions
• Solving trigonometric equations
• Reflection of graphs across y=x