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Trigonometric Functions & the Unit Circle — Visual Lab
Module 3. One point, dragged around a circle of radius 1, is secretly drawing a wave. Sweep the angle and watch the unit circle and the sine curve stay locked together — coordinates, radians, and degrees all in agreement.
The unit circle is the bridge from right-triangle ratios to functions that repeat. Stand an angle \(\theta\) at the center in standard position; the point where its ray meets the circle has coordinates \(\big(\cos\theta,\ \sin\theta\big)\) — cosine is the run, sine is the rise. Spin the angle and the height of that point, plotted against the angle, traces the sine curve. Drive the lab below and keep the circle, the wave, the algebra, and the exact special-angle values in perfect agreement.
Unit Circle ↔ Wave Lab
Drag the terminal point around the circle (or sweep the angle \(\theta\) slider). The right panel live-traces \(y=\sin\theta\) as the angle grows; toggle the cosine wave to see the x-coordinate too. The readout shows the exact \((\cos\theta,\ \sin\theta)\) and the angle in both radians and degrees.
What you're seeing
- The point on the rim is \((\cos\theta,\ \sin\theta\). Its horizontal coordinate is the cosine; its vertical coordinate is the sine. The two dashed legs are exactly those coordinates.
- Angles start at the positive x-axis (standard position) and grow counter-clockwise. The shaded wedge is \(\theta\); the slider reads it in degrees, the readout also in radians.
- The height becomes the wave. Plot the point's height (\(\sin\theta\)) against the angle and you trace the sine curve on the right — that's why it's periodic.
- Reference angle & quadrant. The readout names the quadrant and the acute reference angle — the key to the sign of each coordinate (cosine is the x-sign, sine is the y-sign).
- Special angles are exact. Land on \(30^\circ, 45^\circ, 60^\circ,\dots\) and the readout shows the exact fraction/radical values, not rounded decimals.
Try this
- Sweep slowly from \(0^\circ\) to \(360^\circ\). Watch the sine trace rise to \(1\) at \(90^\circ\), return to \(0\) at \(180^\circ\), dip to \(-1\) at \(270^\circ\), and close at \(360^\circ\). One full lap = one full wave.
- Stop at \(30^\circ, 45^\circ, 60^\circ\). Read the exact coordinates. Notice \(\cos\) and \(\sin\) swap between \(30^\circ\) and \(60^\circ\) — that's the complementary-angle relationship.
- Turn on the cosine wave. See that it is the sine curve shifted left by \(90^\circ\) (\(\tfrac{\pi}{2}\)): \(\cos\theta = \sin\!\big(\theta + 90^\circ\big)\).
- Find every angle where \(\sin\theta = \tfrac12\). Drag until the y-coordinate is \(0.5\). You'll hit \(30^\circ\) and \(150^\circ\) — same reference angle, two quadrants.
Worked Examples
Two of the moves this module asks for most: converting between degrees and radians, and reading an exact coordinate off the unit circle with a reference angle.
Example 1 — Convert \(225^\circ\) to radians, then locate it
- Multiply by \(\dfrac{\pi}{180^\circ}\). \(225^\circ \cdot \dfrac{\pi}{180^\circ} = \dfrac{225\pi}{180}\).
- Reduce the fraction. \(\dfrac{225}{180} = \dfrac{5}{4}\), so \(225^\circ = \dfrac{5\pi}{4}\).
- Place it. \(225^\circ\) is \(180^\circ + 45^\circ\), so it lands in Quadrant III with a reference angle of \(45^\circ\).
- Read the coordinates. The \(45^\circ\) values are \(\tfrac{\sqrt2}{2}\), and both are negative in Quadrant III: \(\big(\cos 225^\circ,\ \sin 225^\circ\big) = \big(-\tfrac{\sqrt2}{2},\ -\tfrac{\sqrt2}{2}\big)\).
Example 2 — Model a Ferris wheel as a sinusoid
A Ferris wheel has a radius of \(20\) m, its center sits \(24\) m above the ground, and it completes one turn every \(40\) seconds. A rider boards at the bottom. Write a height-vs-time model \(h(t)\).
- Amplitude = radius. The car rises and falls \(20\) m from center, so amplitude \(A = 20\).
- Midline = center height. The wheel oscillates around \(24\) m, so the midline is \(k = 24\).
- Period → \(b\). One turn is \(40\) s, so \(b = \dfrac{2\pi}{\text{period}} = \dfrac{2\pi}{40} = \dfrac{\pi}{20}\).
- Start at the bottom. Bottom means start at the minimum, so use a flipped cosine: \(h(t) = -20\cos\!\big(\tfrac{\pi}{20}t\big) + 24\).
Key Vocabulary
The precise words a mathematician uses to describe what the lab is doing.
The circle of radius \(1\) centered at the origin. A point on it at angle \(\theta\) has coordinates \((\cos\theta,\ \sin\theta)\).
An angle drawn with its vertex at the origin and initial side along the positive x-axis; positive angles open counter-clockwise.
An angle measure based on arc length: a full circle is \(2\pi\) radians \(= 360^\circ\), so \(180^\circ = \pi\) rad.
The acute angle between the terminal side and the x-axis. It sets the size of each coordinate; the quadrant sets the sign.
For \(y=a\sin(bx)+k\): amplitude \(|a|\) is half the peak-to-trough height; the midline \(y=k\) is the center the wave oscillates around.
The horizontal length of one full cycle. For \(y=\sin(bx)\) the period is \(\dfrac{2\pi}{|b|}\) — how often the wave repeats.
TEKS & Function Key-Features
This lab targets the Trigonometric Functions strand of Pre-Calculus, where scholars build the unit circle, convert fluently between degrees and radians, use reference angles in standard position, and connect the six trig ratios to periodic graphs.
P.4A determine the value of trig functions using the unit circle and periodicity · P.4B describe radian measure and convert degree↔radian · P.4C use reference angles in standard position · P.4E determine the trig ratios · P.2P determine values of trig functions at special angles · P.2O develop and use sinusoidal models. Standards from 19 TAC §111.42.
Ready for the full course map? Head back to Pre-Calculus, read the Student Support, or check the Pacing Guide to see where Module 3 sits in the year.
Module and topic structure follow the TEA Bluebonnet Learning — Secondary Mathematics open curriculum (Edition 1, adapted from Carnegie Learning), licensed CC BY-NC 4.0. Classroom use is non-commercial.