Trigonometric Functions: Circular Motion

Learning Objectives

By the end of this section you will be able to:

1. Define sine, cosine, tangent and their reciprocals from both the right-triangle picture and the unit-circle picture, and explain why the unit-circle definition is the "true" one.
2. Convert fluently between degrees and radians, and explain why radians are the natural angle unit for calculus.
3. Recognise the four-parameter wave family $y = A\,\sin\bigl(B(x - C)\bigr) + D$ and read off amplitude, period, phase and offset from a graph.
4. Apply the Pythagorean, sum, and double-angle identities to simplify expressions and solve modeling problems.
5. Model periodic phenomena (sound, AC current, tides, body temperature, orbital motion) with sinusoidal functions.
6. Compute trig values in Python and explain why $np.sin$ expects radians, never degrees.

The Story: Where Trigonometry Came From

Trigonometry was invented twice. The first time, around 2,000 years ago, by astronomers (Hipparchus, Ptolemy) who needed to predict where a planet would appear in the night sky. The geometry of a circle was the only way to handle the rotation of the heavens. The second time, in the 17th and 18th centuries, the same functions were rediscovered as the natural way to describe any oscillation: a pendulum, a vibrating string, an AC voltage, a sound wave, a heartbeat.

Why this matters: Anything that repeats — a planet circling the sun, a guitar string vibrating, your circadian rhythm, the alternating current in a wall outlet, the bit pattern of a Wi-Fi signal — speaks the language of sine and cosine. Once you can read this language, periodic motion becomes a four-knob machine you can dial in.

The single most important picture in trigonometry is the unit circle — a circle of radius 1 centered at the origin. Every trig function is just a different way of measuring something about a point walking around that circle.

First Definition: Ratios in a Right Triangle

Drop a perpendicular from the moving point to the x-axis. You get a right triangle whose hypotenuse is the radius (length 1), whose horizontal leg has length $\cos\theta$, and whose vertical leg has length $\sin\theta$. The ancient definitions are the three ratios you can form from the three sides:

The schoolroom mnemonic SOH-CAH-TOA is just these three ratios. They work fine for angles between $0$ and $\pi/2$ (a right triangle cannot have an obtuse angle as the angle of interest). But what about $\theta = 120^\circ$? Or $\theta = -45^\circ$? Or a full rotation of $\theta = 720^\circ$? The triangle picture breaks. We need the unit-circle picture.

Better Definition: The Unit Circle

Place a circle of radius 1 at the origin. Measure an angle $\theta$ counter-clockwise from the positive x-axis. The point where the rotating radius meets the circle has coordinates

That is the definition. Three things become immediate consequences:

1. Pythagorean identity. The point sits on a circle of radius 1, so by the distance formula $\cos^2\theta + \sin^2\theta = 1$. Always.
2. Range. Coordinates on a unit circle live in $[-1, 1]$, so $-1 \le \sin\theta \le 1$ and $-1 \le \cos\theta \le 1$.
3. Periodicity. Going around once returns you to the start, so $\sin(\theta + 2\pi) = \sin\theta$ and the same for cosine.

Why Radians? The Natural Angle Unit

A degree is a human invention: the Babylonians divided a full turn into 360 because 360 has many divisors (and roughly matches the number of days in a year). It is a pragmatic choice with no mathematical justification.

A radian, in contrast, is the angle subtended at the center of a circle by an arc equal in length to the radius. On the unit circle this means:

A full turn traces an arc of length $2\pi$, so $360^\circ = 2\pi$ radians. The conversion is

Special Angles You Must Know

There are five angles whose sine and cosine you should be able to recall in your sleep. They come from the geometry of two simple triangles: the 45-45-90 (half a unit square) and the 30-60-90 (half an equilateral triangle).

All Six Trig Functions

From sine and cosine you can build four more functions by taking ratios and reciprocals. You only need to memorize the two; the rest are derived:

Graphs of Sine and Cosine

If you start a stopwatch and watch the y-coordinate of a point spinning around the unit circle, you trace out the graph of $y = \sin t$. If you watch the x-coordinate, you trace out $y = \cos t$. The animation below makes this concrete.

From this picture, a few facts are visible:

• Both $\sin$ and $\cos$ oscillate between $-1$ and $1$.
• They have the same shape but cosine is shifted left by $\pi/2$: $\cos\theta = \sin\bigl(\theta + \tfrac{\pi}{2}\bigr)$.
• Sine is odd: $\sin(-\theta) = -\sin\theta$. Cosine is even: $\cos(-\theta) = \cos\theta$. (Geometrically: flipping the unit circle over the x-axis sends y to −y but leaves x alone.)

The Four Knobs: A · sin(B(x − C)) + D

Every sine wave that nature can produce — the pressure variation of a pure tone, the voltage at a wall outlet, the height of the tide at a port — fits the same template:

Each letter has a single physical meaning. The order matters because each knob acts on a different axis:

The Identities That Tie It Together

An identity is an equation that holds for every value of the variable, not just some. Trig has dozens, but four families are enough to derive the rest.

1. Pythagorean identities

Direct from $x^2 + y^2 = 1$ on the unit circle:

Dividing through by $\cos^2\theta$ and by $\sin^2\theta$ gives two siblings:

2. Even / odd identities

3. Co-function (complementary-angle) identities

4. Sum and difference formulas

The most useful identities of all. Memorize these two:

Plugging $\alpha = \beta = \theta$ gives the double-angle formulas:

Modeling Periodic Phenomena

Whenever a quantity oscillates between two extremes over a fixed period, the four-knob template gives you a model with very little extra work. The recipe is:

1. Find the midline: $D = \tfrac{\max + \min}{2}$.
2. Find the amplitude: $A = \tfrac{\max - \min}{2}$.
3. Find the angular frequency: $B = \tfrac{2\pi}{T}$, where $T$ is the observed period.
4. Find the phase shift $C$ from where the peak (for cosine) or zero-crossing-going-up (for sine) actually occurs.

Pattern to remember: the shape of the wave (sine vs cosine, which phase) is always a stylistic choice. The four numbers A, B, C, D carry all the physics. Find them, write the formula, you are done.

Sums of Sines: A Peek at Fourier

A single sine wave is monotonous. Reality — a violin note, a spoken vowel, a radio signal — is more complicated. The Fourier theorem says something astonishing: any periodic function (under very mild conditions) is a sum of sines and cosines with different amplitudes and frequencies.

Play with the sliders. Three sine waves are already enough to make complex, irregular-looking shapes. A full violin tone uses dozens. An MP3 file is basically a list of amplitudes for thousands of these sine waves. The whole field of signal processing exists to find clever ways of computing and storing those amplitudes.

Computing with Python and NumPy

Now let us see how all this looks in code. We start with plain Python to keep the algorithm visible, then move to NumPy where vectorization makes everything fast. Both libraries use the radian convention — feeding them degrees by mistake is one of the most common bugs in scientific Python.

1. Plain Python: building a sine table by hand

1import math
2
3# Five "special" angles in radians, paired with their fraction-of-pi labels.
4special_angles = [
5    (0.0,            "0"),
6    (math.pi / 6,    "pi/6"),
7    (math.pi / 4,    "pi/4"),
8    (math.pi / 3,    "pi/3"),
9    (math.pi / 2,    "pi/2"),
10]
11
12print(f"{'angle':>8} | {'rad':>8} | {'sin':>10} | {'cos':>10}")
13print("-" * 46)
14
15for theta, label in special_angles:
16    s = math.sin(theta)
17    c = math.cos(theta)
18    print(f"{label:>8} | {theta:>8.4f} | {s:>10.6f} | {c:>10.6f}")

The expected output is exactly the textbook table:

   angle |      rad |        sin |        cos
----------------------------------------------
       0 |   0.0000 |   0.000000 |   1.000000
    pi/6 |   0.5236 |   0.500000 |   0.866025
    pi/4 |   0.7854 |   0.707107 |   0.707107
    pi/3 |   1.0472 |   0.866025 |   0.500000
    pi/2 |   1.5708 |   1.000000 |   0.000000

2. NumPy: vectorized version for plotting

Plain Python is great for one value at a time. For the thousands of points you need to draw a sine wave, NumPy is 100× faster because the whole loop runs in compiled C under the hood.

1import numpy as np
2import matplotlib.pyplot as plt
3
4# 1) Build 500 evenly spaced sample points over two full periods (0 to 4π).
5t = np.linspace(0.0, 4.0 * np.pi, 500)
6
7# 2) Choose the four wave knobs: amplitude, angular frequency, phase, offset.
8A, B, C, D = 2.0, 1.5, 0.5, 0.3
9
10# 3) Vectorized formula — runs once on the WHOLE array, no Python loop.
11y = A * np.sin(B * (t - C)) + D
12
13# 4) Compute period and frequency from B.
14T = 2.0 * np.pi / B
15f = 1.0 / T
16print(f"period T = {T:.4f} s, frequency f = {f:.4f} Hz")
17
18# 5) Quick plot — set up a clean figure.
19fig, ax = plt.subplots(figsize=(8, 3))
20ax.plot(t, y, color="#22d3ee", linewidth=2.2)
21ax.axhline(D, color="#a78bfa", linestyle="--", linewidth=1)
22ax.set_xlabel("t (radians)")
23ax.set_ylabel("y")
24ax.set_title("y = A sin(B(t-C)) + D")
25ax.grid(alpha=0.3)
26plt.show()

Expected console output:

period T = 4.1888 s, frequency f = 0.2387 Hz

3. Verifying the Pythagorean identity numerically

1import numpy as np
2
3# Generate one million random angles in the range [-10π, 10π].
4rng = np.random.default_rng(seed=42)
5theta = rng.uniform(-10.0 * np.pi, 10.0 * np.pi, size=1_000_000)
6
7# Pythagorean identity, vectorized.
8lhs = np.sin(theta) ** 2 + np.cos(theta) ** 2
9
10# How far from 1 do we ever get?
11max_error = np.max(np.abs(lhs - 1.0))
12mean_error = np.mean(np.abs(lhs - 1.0))
13print(f"max absolute error  = {max_error:.3e}")
14print(f"mean absolute error = {mean_error:.3e}")

Expected output:

max absolute error  = 2.220e-16
mean absolute error = 5.551e-17

Calculus Preview: Why sin and cos Are Magical

We have not done calculus yet, but you should know what makes sin and cos uniquely beautiful before we get there.

The miracle of trigonometry: $\dfrac{d}{d\theta}\sin\theta = \cos\theta, \quad \dfrac{d}{d\theta}\cos\theta = -\sin\theta.$

That is to say: the rate of change of sin is cos, and the rate of change of cos is −sin. This is the only pair of functions (besides exponentials) that is essentially its own derivative. And this is precisely why every linear differential equation describing oscillation — a mass on a spring, a pendulum, an LC circuit, a photon — has sines and cosines as solutions.

We will derive both of these formulas from the limit definition in Chapter 5. The key ingredient turns out to be a special limit you will meet in Chapter 2: $\lim_{h \to 0} \tfrac{\sin h}{h} = 1$ — the statement that for small angles, sine and angle are the same number.

Summary

1. Definition. On the unit circle a point at angle θ has coordinates $(\cos\theta, \sin\theta)$. Every trig function is a ratio or reciprocal built from these two.
2. Radians. Angle = arc length on the unit circle. One full turn is $2\pi$ radians. All of calculus assumes this unit.
3. Special angles. 0, π/6, π/4, π/3, π/2 give the table you should know by heart.
4. Graphs. $y = A\sin(B(x-C)) + D$ covers every sinusoid you will ever see: amplitude A, period 2π/B, phase shift C, midline D.
5. Identities. Pythagorean ($\sin^2 + \cos^2 = 1$), even/odd, complementary, sum/difference, double-angle. Each is a tool for converting one form to a more useful one.
6. Modeling. Periodic phenomena follow the four-knob template; find midline, amplitude, period, phase from the data.
7. Fourier hint. Every periodic signal is a sum of sines. We will formalise this much later.
8. Code. $math.sin$ and $np.sin$ both expect radians. Mixing units silently is the #1 trig bug.

Exercises

Conceptual

1. Explain in one sentence, using the unit-circle picture, why $\sin(\pi + \theta) = -\sin\theta$.
2. Why is $\tan(\pi/2)$ undefined, but $\sin(\pi/2)$ is perfectly fine? Answer with coordinates, not just "division by zero".
3. Without computing, decide whether $\cos(5\pi/3)$ is positive or negative. Which quadrant?

Computational

1. Convert 210° to radians and find $\sin(210^\circ)$ and $\cos(210^\circ)$ exactly.
2. Find amplitude, period, phase shift, and midline of $y = -3\cos\!\bigl(\tfrac{\pi}{2}(x + 1)\bigr) + 4$.
3. Use the sum formula to find $\cos(15^\circ)$ exactly. Hint: 15 = 45 − 30.
4. Solve $2\sin\theta - 1 = 0$ for all $\theta \in [0, 2\pi)$.

Modeling

1. A Ferris wheel of radius 20 m turns once every 90 s. A passenger starts at the bottom (3 m above ground). Write a function for the passenger's height vs time. What is the height at $t = 30$ s?
2. A 60 Hz alternating-current outlet supplies voltage $V(t) = 170\sin(120\pi t)$. What are amplitude, period, and frequency? Why is "120π" the angular frequency for a 60 Hz signal?
3. Daylight in Boston ranges from 9.0 h (Dec 21) to 15.3 h (Jun 21). Build a sinusoidal model $L(d)$ for daylight hours as a function of day-of-year $d$. How many hours of daylight on March 21 (day 80)?

Programming

1. Write a Python function $degrees_to_radians(d)$ from scratch (no $math.radians$). Verify with at least three test angles.
2. Plot $f(x) = \sin(x) + \tfrac{1}{3}\sin(3x) + \tfrac{1}{5}\sin(5x)$ on $[-\pi, \pi]$. What familiar shape does this look like? (Hint: it converges to a square wave.)
3. Vectorize the Pythagorean-identity check from this section to also verify $1 + \tan^2 = \sec^2$. Be careful around $\theta = \pi/2 + k\pi$ where tan blows up.

In the next section we will invert these functions — arcsin, arccos, arctan — to answer the reverse question: "given a sine value, what angle produced it?" That requires choosing a single branch and is the cousin of the multi-valued conversation we began here.