Unroll the spinning point into an endlessly repeating wave.
In the unit circle we pinned the spinning point down: at angle θ it sits at P(cos θ, sin θ), so its height is sin θ and its across is cos θ. Now let the angle just keep running — past 360°, around again, on and on — and watch only the height. Plot that height against the angle and a brand-new picture appears: a smooth, endlessly repeating wave. This is where trigonometry stops being about triangles and becomes the language of everything that cycles — a heartbeat, a sound, a tide, the daylight across a year. In this lesson we unroll the circle into the sine wave, meet its twin the cosine, read off their period, range, and symmetry, find where they climb and fall, and finally meet the wilder tangent graph that races off to infinity.
Set the unit circle on the left and an empty x-axis on the right, where x will measure the angle (in radians). Start the point at θ = 0: it sits at (1, 0), height 0. Now turn it counter-clockwise. As θ grows the point climbs; its height is exactly sin θ. At each moment, carry that height straight across to x = θ and drop a dot. String the dots together and you are drawing y = sin x.
Watch the four landmarks. At θ = π/2 the point is at the very top, (0, 1): the curve reaches its crest +1. At θ = π the point is back on the axis at (−1, 0): height 0 again. At θ = 3π/2 the point hangs at the bottom, (0, −1): the curve bottoms out at −1. At θ = 2π we are home at (1, 0), height 0 — one full arch-and-valley complete. Keep turning and the whole shape repeats, because the point is literally back where it started.
The sine graph is the unit circle's height, unrolled. Reading sin x for an angle and reading the y-coordinate of the point at that angle are the same act — the wave is just the circle stretched out straight.
Dial θ. The point on the left climbs and falls; the wave on the right grows to match, with a green dot carrying the current height across.
The cosine is born the same way — but it tracks the point's across, its x-coordinate, instead of its height. Start again at θ = 0: now the point is at (1, 0), so cos 0 = 1 — the cosine wave starts at the top. As θ runs it falls to 0 at π/2, to −1 at π, and back. Same smooth wave, same range, same length — just begun a quarter-turn earlier.
Both curves have period 2π — the smallest horizontal distance after which the picture exactly repeats — and both have range [−1, 1]: the height of a point on the unit circle can never exceed 1 or drop below −1, so neither wave ever escapes that band. Because the cosine is the sine shifted, the two formulas tie together: cos x = sin(x + π/2) and sin x = cos(x − π/2).
Where does y = cos x cross zero? Wherever the point's across is 0 — at the top and bottom of the circle, θ = π/2 and 3π/2, and every half-turn after: x = π/2 + kπ. The sine crosses zero a quarter-turn earlier, at x = 0, π, 2π, … = kπ.
Toggle sin or cos, then slide x. The readout names the period, the range, and whether the curve is climbing or falling there.
That "repeats every 2π" is the deepest fact about these functions, and it has a name: they are periodic, with period 2π. Formally, sin(x + 2π) = sin x and cos(x + 2π) = cos x for every x — adding a full turn lands the point on the exact same spot, so it has the exact same height and across.
The two waves also have opposite symmetry. Reflecting the angle, θ → −θ, flips the point above-or-below the x-axis: the height reverses sign but the across is unchanged. So sin(−x) = −sin x (sine is odd — its graph has half-turn symmetry about the origin O) while cos(−x) = cos x (cosine is even — its graph is mirror-symmetric across the y-axis). You can see it: fold the sine graph through O and it lands on itself; fold the cosine graph across the y-axis and it lands on itself.
Odd vs even is just θ → −θ on the circle. Flipping the angle flips the height (sin reverses) but keeps the across (cos unchanged). That single picture explains both symmetries at once.
Between its crest (max +1) and its trough (min −1) each wave is monotonic — it only climbs, or only falls. For y = sin x over one period starting at 0: it increases on (−π/2, π/2), reaching the crest +1 at π/2; then it decreases on (π/2, 3π/2), bottoming at −1 at 3π/2; then climbs again. Because of the 2π repeat, that pattern of "up a quarter, down a half, up a quarter" tiles the whole axis.
The cosine, shifted a quarter-turn, hits its crest +1 at x = 0 (and every 2π), its trough −1 at x = π, and increases on (π, 2π), decreases on (0, π). The extremes never move outside [−1, 1] — that is the whole point of "bounded."
The wave never leaves [−1, 1]. If a calculation hands you "sin x = 1.4," something is wrong — no angle has a height bigger than the radius of a unit circle. (The tangent we meet next is a different story.)
Tangent is the ratio tan x = sin xcos x — height divided by across. Two things follow immediately. First, it is 0 wherever sin x = 0, at x = 0, π, 2π, … Second, and dramatically, it blows up wherever cos x = 0 — at x = π/2 + kπ — because you are dividing by zero. There the curve has a vertical asymptote: as x creeps toward π/2 from the left, tan x rockets to +∞; just past it, it returns from −∞.
Notice the period is π, not 2π: adding half a turn flips both sin x and cos x in sign, and a ratio of two flipped signs is unchanged, so tan(x + π) = tan x. Like sine, tangent is odd: tan(−x) = −tan x.
Slide x toward π/2. Watch tan x explode while sin and cos stay politely inside [−1, 1].
tan x is undefined, not "infinity," at 90°, 270°, … There is no point on the graph there — the dashed red line is a wall the curve hugs but never touches.
Three waves, three personalities. Sine and cosine are bounded twins of period 2π; tangent is the unbounded one of period π that lives between asymptotes.
| function | period | range | parity | symmetry | special features |
|---|---|---|---|---|---|
| y = sin x | 2π | [−1, 1] | odd | about O | crest +1 at π/2, trough −1 at 3π/2; zero at kπ |
| y = cos x | 2π | [−1, 1] | even | about y-axis | crest +1 at 0, trough −1 at π; cos x = sin(x + π/2) |
| y = tan x | π | (−∞, ∞) | odd | about O | asymptotes at π/2 + kπ; zero at kπ |
Next, in sinusoidal functions, we take this plain sine wave and add three dials — amplitude, frequency, phase — to model any vibration. But first, trig identities builds the algebra that ties all these curves together.
What is the period and the range of y = sin x?
Period 2π (it repeats after one full turn of the circle) and range [−1, 1] (the point's height never exceeds the unit radius).
Use the unit circle to evaluate sin(−π/2) and explain which symmetry you are using.
sin(−π/2) = −1. Reflecting the angle to −θ flips the point below the axis, reversing the height: sin(−x) = −sin x (sine is odd). Since sin(π/2) = 1, sin(−π/2) = −1.
On the interval (0, 2π), where is y = cos x increasing?
Cosine bottoms out at −1 at x = π, then climbs back to +1 at x = 2π, so it is increasing on (π, 2π). (It is decreasing on (0, π).)
Why does y = tan x have a vertical asymptote at x = π/2, and what is its period?
Because tan x = sin x / cos x and cos(π/2) = 0, so we divide by zero — tan x is undefined there and races to ±∞. Adding π flips both sin and cos, leaving the ratio unchanged, so the period is π.
A classmate writes "cos x = sin(x − π/2)." Is the shift correct?
No — it should be cos x = sin(x + π/2). Cosine starts at its crest (+1 at x = 0), which is sine shifted left by π/2. The correct pair is cos x = sin(x + π/2) and sin x = cos(x − π/2).
Without a calculator, decide which of these can be the value of a sine: 0.3, −1, 1.4, −0.99. Explain.
0.3, −1, and −0.99 are all fine (they lie in [−1, 1]). 1.4 is impossible — sine is bounded by the unit radius, so no angle has sin x > 1.
Six questions to lock it in. Tap the answer you think is right.
This lesson covers the graphs and properties of the trigonometric functions, aligned with CCSS HSF-TF.B.5 (modeling periodic phenomena), HSF-TF.A.4 (using the unit circle to explain symmetry and periodicity of sine, cosine, and tangent), and HSF-IF.B.4 / IF.C.7e (interpreting and graphing periodic functions, identifying period, midline, amplitude, and end behavior). The "unrolling the circle" widget makes the link between a rotation and its wave concrete; the tangent widget shows graphically why dividing by cos x = 0 produces a vertical asymptote rather than a value.