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

This page assumes nothing. Before you touch the parent note the graphs of sin, cos and tan, we build every symbol it quietly relies on, in an order where each piece rests on the one before it.

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1. What is an angle, and why radians?

Picture a clock hand pinned at the centre. Swing it from pointing-right up to pointing-up: the amount of that swing is the angle. Turn is turn — it does not care how long the hand is.

We can measure that turn two ways:

• Degrees: split one full turn into $360$ equal slices. One slice is $1^{\circ }$.
• Radians: measure the turn by how far you walked along the rim of a circle of radius $1$.

WHY radians and not degrees? Because later the topic wants the slope and rate of these waves to come out clean. When angle is measured as arc-length, the sine wave's steepness at the origin is exactly $1$ — no ugly conversion factor of $\pi /180$ floating around. Radians make the geometry and the calculus speak the same language. This is why the parent writes $2\pi$ for a full period, not $360$.

Full radian detail lives in Unit Circle and Radian Measure.

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2. Coordinates: reading a point's address $(x,y)$

Picture graph paper. Start at the centre, walk $x$ steps right, then $y$ steps up — you land on the point. That pair of numbers is all you need to name it.

WHY the topic needs this: the parent says a point on the circle is $(\cos x,\sin x)$. That is literally its address. $\cos$ is the sideways number, $\sin$ is the up number. No coordinates → those symbols are meaningless.

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3. The unit circle — the machine that makes everything

Now the master move. Take the angle $x$ from Section 1 and swing our radius-$1$ hand anticlockwise by that much, starting from pointing-right. The tip lands somewhere on the rim. That tip has an address $(x\text{-coord},y\text{-coord})$ from Section 2.

That is the entire definition. Everything in the parent note is squeezing consequences out of this.

WHY the range is $[-1,1]$: the tip lives on the rim, distance $1$ from centre. You can be at most $1$ to the right, at most $1$ up. So neither coordinate ever leaves $[-1,1]$. The wave's ceiling and floor are the circle's edges — nothing more mysterious than that.

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4. The ratio $\frac{\text{something}}{\text{something}}$ — and dividing by zero

The parent writes $\tan x=\frac{\sin x}{\cos x}$. Two symbols to earn: the fraction bar and the danger of dividing by zero.

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5. Positive and negative — the four quadrants

The circle's rim splits into four quadrants. Knowing the sign of each coordinate in each quadrant is the secret behind the wave rising and falling.

Read it off the picture: in quadrant II the tip is up-and-left, so it is up (positive $\sin$) but left of centre (negative $\cos$). That single sign-flip is why cosine crosses zero and dips negative while sine is still high. The parent's whole wave shape is this table, unrolled.

The degenerate corners (where an axis is crossed exactly) matter too:

• At $x=0$: tip at $(1,0)$ → $\cos =1,\sin =0$.
• At $x=\frac{\pi }{2}$: tip at $(0,1)$ → $\cos =0,\sin =1$. Here $\cos =0$, so $\tan$ blows up.
• At $x=\pi$: tip at $(-1,0)$ → $\cos =-1,\sin =0$.
• At $x=\frac{3\pi }{2}$: tip at $(0,-1)$ → $\cos =0,\sin =-1$. Again $\tan$ undefined.

These exact corner values come from Exact Values of Trig Ratios.

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6. Repetition — what "period" and $f(x+P)=f(x)$ really say

The picture: walk once fully around the circle — a jump of $2\pi$ — and you are back at the same tip, so the same height and sideways. Hence $\sin$ and $\cos$ repeat every $2\pi$. That is where the parent's period $=2\pi$ is born. (Tangent, being a ratio, sneakily repeats after only $\pi$ — the parent proves this once you have the signs from Section 5.)

The symbol $n$ (as in "zeros at $n\pi$") just means "any whole number" $n=0,\pm 1,\pm 2,\dots$ — a shorthand for "keep adding full jumps forever."

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7. Reading a graph — axes, amplitude, midline

When we unroll the circle, we plot angle $x$ along the horizontal and the height $\sin x$ up the vertical. The circular motion becomes a wave.

The midline is the horizontal line halfway between top and bottom (here $y=0$). These are the words the parent's transformation section ($A,B,C,D$) manipulates — see Transformations of Graphs.

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8. Prerequisite map

Everything on the left is what this page built; the single node on the far right is the parent topic. Nothing enters the parent that we did not first earn here.

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Equipment checklist

Test yourself — you are ready for the parent note only if every answer comes instantly.

Connections

• Unit Circle and Radian Measure — the machine and its angle-measure
• Exact Values of Trig Ratios — the corner values $0,\frac{\pi }{2},\pi ,\dots$
• Trigonometric Identities — where $\tan =\sin /\cos$ becomes a tool
• Transformations of Graphs — uses amplitude, midline, period
• Solving Trigonometric Equations — leans on periodicity and $n$
• Simple Harmonic Motion — physics reading of the unrolled wave