Visual walkthrough — Graphs of sin x, cos x, tan x — key features, period, amplitude

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Step 1 — Set up the stage: the unit circle

WHAT. Draw a circle whose radius is exactly $1$. Put a dot on its right edge. We call the point where the dot sits $P$. The distance from the centre $O$ to $P$ is $1$ — that is what "unit" means (unit = one).

WHY. Everything about these waves is a ratio or a length measured against radius $1$. Choosing radius $1$ makes all those numbers come out clean: a length of "half the radius" is just the number $0.5$. See Unit Circle and Radian Measure for why radius $1$ is the natural yardstick.

PICTURE. Look at the figure. The horizontal line is the $x$-axis, the vertical line is the $y$-axis, and the dot $P$ starts at the far right, at coordinates $(1,0)$.

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Step 2 — Name the two shadows: $\cos x$ and $\sin x$

WHAT. Drop the dot $P$ straight down onto the $x$-axis — the length of that horizontal reach is what we call $\cos x$. Now slide $P$ straight across onto the $y$-axis — that vertical reach is $\sin x$. So the dot's coordinates are exactly $(\cos x,\sin x)$.

WHY these two and not something else? A point in a plane needs two numbers to pin it down: how far right, how far up. Those two numbers are cosine and sine. They are not extra machinery bolted on — they are literally the point's address.

PICTURE. In the figure, the blue horizontal segment is $\cos x$ and the yellow vertical segment is $\sin x$. Together with the radius $OP$ they form a right triangle (a triangle with one square corner). The radius is the sloped side of length $1$.

Because the slanted side is $1$ and the other two sides can never be longer than it, neither shadow can exceed $1$ in size. That single sentence is where the range $[-1,1]$ comes from.

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Step 3 — Unroll the height: sine becomes a wave

WHAT. Keep the circle on the left. To its right, lay out a horizontal axis labelled $x$ (the angle). For each position of the dot, carry its height $\sin x$ straight across and plot it at horizontal position $x$. As the dot walks round, these carried-over heights trace a curve.

WHY. We want a graph of "height versus angle." The circle already knows the height at every angle; we are just re-plotting that same information on a flat ruler so we can see the whole story at once instead of one frozen dot.

PICTURE. The green dashed line carries the yellow height across from the circle to the curve. Watch what the height does over one lap:

• At $x=0$: dot at the right, height $0$ → curve starts at sea level.
• At $x=\frac{\pi }{2}$: dot at the top, height $1$ → curve at its peak.
• At $x=\pi$: dot at the left, height $0$ → curve back to sea level.
• At $x=\frac{3\pi }{2}$: dot at the bottom, height $-1$ → curve at its trough.
• At $x=2\pi$: dot back at the right, height $0$ → one full wave done.

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Step 4 — Unroll the width: cosine is the same wave, shifted

WHAT. Do the identical unrolling, but this time carry across the horizontal shadow $\cos x$ instead of the height. (To carry a horizontal length onto a vertical graph, imagine tipping it up.) Plot $\cos x$ against $x$.

WHY. Cosine deserves its own graph because it answers a different question — "how far right is the dot?" — and that question peaks at a different time than the height does.

PICTURE. The blue curve is $\cos x$. Compare it against the faint yellow sine curve behind it:

• At $x=0$: dot fully right, $\cos 0=1$ → cosine starts at its peak (sine started at zero).
• At $x=\frac{\pi }{2}$: dot at top, no rightward reach, $\cos =0$.
• At $x=\pi$: dot fully left, $\cos =-1$ → trough.

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Step 5 — Build tan as a ratio: slope of the arm

WHAT. Define a third quantity: divide the height by the width. $\tan x=\frac{\sin x}{\cos x}=\frac{\text{how far up}}{\text{how far right}}.$ This is exactly the slope (steepness) of the line $OP$ — rise over run. See Trigonometric Identities for this ratio as an identity.

WHY a ratio, and why this ratio? "Slope" answers "how tilted is the arm?" — a single number for the whole direction. Height alone or width alone can't tell you tilt; only their ratio can, because a steep line has lots of rise per unit run.

PICTURE. The red line is $OP$ extended. The little triangle shows rise $=\sin x$ and run $=\cos x$; their ratio is the steepness. When the arm is gently sloped, $\tan x$ is small; when it points nearly straight up, run shrinks toward $0$ and the ratio blows up.

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Step 6 — The cliffs: where tan explodes (asymptotes)

WHAT. Unroll $\tan x$ into a graph the same way, but now mark the danger points. $\cos x=0$ happens when the dot is at the top ($x=\frac{\pi }{2}$) or bottom ($x=\frac{3\pi }{2}$) of the circle — the arm points straight up or straight down, so its run is zero.

WHY vertical lines appear. Just before the top, run is tiny and positive, so $\frac{\text{rise}}{\text{tiny}}$ is a huge positive number → the curve races up to $+\infty$. Just after the top, run is tiny and negative, so the ratio is a huge negative number → the curve comes up from $-\infty$ on the other side. The graph can never touch these lines; they are vertical asymptotes.

PICTURE. The red dashed vertical lines sit at $x=\frac{\pi }{2}$ and $x=\frac{3\pi }{2}$. Notice each branch of $\tan$ climbs from $-\infty$ up through $0$ and races to $+\infty$, then restarts.

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Step 7 — The edge case: why tan repeats after only $\pi$

WHAT. Move the dot half a lap, i.e. replace $x$ by $x+\pi$. The dot lands on the exact opposite side of the circle. Both shadows flip sign at once: $\sin (x+\pi )=-\sin x,\cos (x+\pi )=-\cos x.$

WHY this is special. Sine alone repeats only after a full lap $2\pi$ (half a lap gives you the negative, not the same value). But $\tan$ is a ratio, and a ratio doesn't care if both parts flip — the minus signs cancel: $\tan (x+\pi )=\frac{-\sin x}{-\cos x}=\frac{\sin x}{\cos x}=\tan x.$

PICTURE. On the left, the arm at angle $x$ and the arm at $x+\pi$ point in opposite directions but lie on the same straight line — so they have the same slope. Same slope = same $\tan$. On the right, the tan graph shows this: one branch is an exact copy of the previous branch, spaced only $\pi$ apart.

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Step 8 — Degenerate check: the exact peak, trough, and zero

WHAT. Verify the "corner" angles by hand so no case is left unexplored.

PICTURE / TABLE. Each row is a dot position we can see, not compute:

$x$	dot at	$\cos x$	$\sin x$	$\tan x=\frac{\sin }{\cos }$
$0$	right $(1,0)$	$1$	$0$	$0/1=0$
$\frac{\pi }{2}$	top $(0,1)$	$0$	$1$	$1/0=$ undefined (asymptote)
$\pi$	left $(-1,0)$	$-1$	$0$	$0/(-1)=0$
$\frac{3\pi }{2}$	bottom $(0,-1)$	$0$	$-1$	$-1/0=$ undefined
$2\pi$	right again	$1$	$0$	$0$

WHY include this. The zero-over-something case ($\tan =0$) and the something-over-zero case ($\tan$ undefined) are the two extremes; seeing both confirms our graph never surprises us. Use these when Solving Trigonometric Equations.

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The one-picture summary

Everything on this page is one dot, one circle, three carried-across measurements. The final figure stacks all three curves under the circle so you can trace each one back to a shadow. When the dot is at the top: sine peaks, cosine is zero, tan explodes. When at the right: sine is zero, cosine peaks, tan is zero. Read it as one story told three ways.

Recall Feynman retelling — explain the whole walkthrough to a friend

Picture a ladybird crawling anticlockwise around a clock face of radius one. We track three things about her:

How high she is (her height above the middle) — plot that against the angle she's walked, and you get a smooth up-and-down wave that starts at zero, peaks a quarter of the way round, dips to a trough three-quarters round, and repeats every full lap. That's sine.

How far right she is — the same wave, but it starts at the top because she begins on the far right. That's cosine, sine given a quarter-lap head start.

Her height divided by her sideways position — that's the steepness of the line from the centre out to her, and that's tangent. Two things go wrong-and-right here: when she's dead level with the centre (straight up or straight down), her sideways position is zero, and dividing by zero makes tan shoot off to a cliff — those are the asymptotes. And when she reaches the exact opposite side of the clock, both her height and her sideways flip sign together, but since tan divides one by the other, the two minus signs cancel and tan looks identical — that's why tan repeats twice as often as the other two, every half-lap $\pi$ instead of $2\pi$.

One bug, one circle, three shadows. That's the entire chapter.

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Connections

• Unit Circle and Radian Measure — the circle every step is built on
• Exact Values of Trig Ratios — the corner values checked in Step 8
• Trigonometric Identities — the ratio $\tan =\sin /\cos$ used in Steps 5–7
• Transformations of Graphs — the quarter-turn shift of Step 4
• Solving Trigonometric Equations — uses the zeros and asymptotes found here
• Simple Harmonic Motion — the physics that runs on these same waves