AP Precalculus Unit 3 Trigonometric Functions: Understanding Cycles, Graphs, and Models

Periodic Phenomena and the Unit Circle

What “periodic” means and why it matters

A periodic phenomenon is anything that repeats in a predictable cycle. The key idea is repetition: after some fixed amount of input, the output returns to the same value and the pattern starts over. That fixed repeat length is the period.

This matters because a huge range of real situations are periodic—day/night temperature, tides, seasonal daylight hours, vibrations, rotating objects, sound waves. Trigonometric functions are built specifically to describe circular motion and repeating patterns, so they become a powerful modeling tool whenever a situation “cycles.”

Mathematically, a function $f$ is **periodic** if there exists a positive number $P$ such that

$f(x+P)=f(x)$

for all $x$ in the domain. The smallest such positive $P$ (if it exists) is called the fundamental period.

A common misconception: students sometimes think “periodic” means “increasing then decreasing.” That shape happens in many periodic graphs, but the definition is about repeating exactly after a fixed horizontal shift.

Angle measurement: degrees and radians

To connect periodic behavior to trig, you need a way to measure rotation.

• Degrees: a full rotation is $360$ degrees.
• Radians: a full rotation is $2\pi$ radians.

Radians are not an arbitrary unit—they directly connect angles to arc length. If a circle has radius $r$, and an angle $\theta$ (in radians) subtends an arc of length $s$, then:

$s=r\theta$

This formula is why radians are the “natural” unit for trigonometry: the angle measure becomes a direct multiplier for arc length. A frequent error is trying to use $s=r\theta$ when $\theta$ is in degrees. The formula only works as written when $\theta$ is in radians.

Conversions follow from the equivalence $360$ degrees $=2\pi$ radians:

$180=\pi$

So:

$\theta_{\text{rad}}=\theta_{\text{deg}}\cdot \frac{\pi}{180}$

$\theta_{\text{deg}}=\theta_{\text{rad}}\cdot \frac{180}{\pi}$

The unit circle as the foundation of trig

The unit circle is the circle of radius $1$ centered at the origin. Its equation is:

$x^2+y^2=1$

The unit circle matters because it gives a clean geometric definition of sine and cosine. If you start at the point $\left(1,0\right)$ and rotate counterclockwise by an angle $\theta$, you land at a point on the unit circle. Call that point $\left(x,y\right)$. Then we define:

• Cosine as the $x$-coordinate
• Sine as the $y$-coordinate

So:

$\cos(\theta)=x$

$\sin(\theta)=y$

This definition automatically builds in periodicity because rotating by $2\pi$ radians brings you back to the same point.

Reference angles and exact values

To evaluate trig functions at many angles, you rely on reference angles: the acute angle formed between the terminal side of $\theta$ and the $x$-axis. The unit circle is symmetric in the four quadrants, so once you know the “special triangle” values in Quadrant I, you can get values elsewhere by assigning signs.

Common special angles (in radians) are:

• $0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}$

A helpful way to remember sine and cosine at these is that they come from the $30-60-90$ and $45-45-90$ triangles scaled to the unit circle.

Here is a compact exact-value table for Quadrant I:

To extend beyond Quadrant I, keep the same reference-angle magnitudes but adjust signs based on the quadrant. A common mistake is mixing up which coordinate corresponds to sine vs cosine—remember: cosine is horizontal (x), sine is vertical (y).

Worked example: arc length on a circle

A bicycle wheel has radius $0.35$ meters. If it rotates through an angle of $\frac{5\pi}{3}$ radians, how far does a point on the rim travel?

Use $s=r\theta$:

$s=0.35\cdot \frac{5\pi}{3}$

$s=\frac{1.75\pi}{3}$

This is an exact distance in meters. If you approximate, you can compute a decimal, but many AP-style questions accept the exact expression.

Exam Focus

• Typical question patterns:
  • Convert between degree and radian measures, often tied to interpreting a graph or rotation.
  • Use the unit circle to find exact values of $\sin(\theta)$ and $\cos(\theta)$ at special angles (including negative angles and angles beyond $2\pi$).
  • Use $s=r\theta$ to connect rotation to distance traveled.
• Common mistakes:
  • Using $s=r\theta$ with $\theta$ in degrees (always convert to radians first).
  • Swapping sine and cosine coordinates on the unit circle.
  • Forgetting sign changes by quadrant when using reference angles.

Sine, Cosine, and Tangent Functions

Defining the trig functions from the unit circle

Using the unit circle point $\left(x,y\right)$ corresponding to angle $\theta$:

$\cos(\theta)=x$

$\sin(\theta)=y$

The tangent function is defined (when cosine is not zero) as the ratio of sine to cosine:

$\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}$

Geometrically on the unit circle, since $\sin(\theta)=y$ and $\cos(\theta)=x$, you can interpret tangent as:

$\tan(\theta)=\frac{y}{x}$

This makes tangent strongly connected to slope: for an angle $\theta$ in standard position, $\tan(\theta)$ matches the slope of the terminal ray (except where the ray is vertical).

Domain, range, and key periodic properties

Understanding how these functions behave globally helps you interpret graphs and models.

Sine:

• Domain: all real numbers
• Range:

$-1\le \sin(\theta)\le 1$

• Period:

$\sin(\theta+2\pi)=\sin(\theta)$

Cosine:

• Domain: all real numbers
• Range:

$-1\le \cos(\theta)\le 1$

• Period:

$\cos(\theta+2\pi)=\cos(\theta)$

Tangent:

• Domain excludes where $\cos(\theta)=0$, which occurs at odd multiples of $\frac{\pi}{2}$
• Range: all real numbers
• Period:

$\tan(\theta+\pi)=\tan(\theta)$

A common misconception: students assume all trig functions have period $2\pi$. Tangent repeats every $\pi$ because rotating by $\pi$ points you in the opposite direction with the same slope.

Even/odd symmetry (helps with negative angles)

Symmetry reduces memorization and helps check answers.

• Cosine is an even function:

$\cos(-\theta)=\cos(\theta)$

• Sine is an odd function:

$\sin(-\theta)=-\sin(\theta)$

• Tangent is an odd function:

$\tan(-\theta)=-\tan(\theta)$

These are easy to remember by thinking about the unit circle: reflecting across the $x$-axis changes the sign of $y$ (sine) but not $x$ (cosine).

Connecting right-triangle trig to unit-circle trig

In many earlier courses, sine, cosine, and tangent are defined in a right triangle as ratios of side lengths. That definition works when $\theta$ is acute. The unit-circle definition extends trig to all real angles (including negative angles and angles greater than $90$ degrees), which is essential for modeling periodic motion.

The triangle ratios still match unit-circle trig for acute angles: if you draw a right triangle inside the unit circle, the coordinates correspond to adjacent/opposite relationships.

Worked example: exact trig values using symmetry and the unit circle

Find exact values of $\sin\left(-\frac{5\pi}{6}\right)$, $\cos\left(-\frac{5\pi}{6}\right)$, and $\tan\left(-\frac{5\pi}{6}\right)$.

1) Start with the positive angle $\frac{5\pi}{6}$. Its reference angle is $\frac{\pi}{6}$, and it lies in Quadrant II.

For $\frac{\pi}{6}$, the Quadrant I values are:

$\sin\left(\frac{\pi}{6}\right)=\frac{1}{2}$

$\cos\left(\frac{\pi}{6}\right)=\frac{\sqrt{3}}{2}$

In Quadrant II, sine is positive and cosine is negative:

$\sin\left(\frac{5\pi}{6}\right)=\frac{1}{2}$

$\cos\left(\frac{5\pi}{6}\right)=-\frac{\sqrt{3}}{2}$

Then tangent:

$\tan\left(\frac{5\pi}{6}\right)=\frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}}=-\frac{1}{\sqrt{3}}$

2) Now use symmetry for the negative angle.

$\sin\left(-\frac{5\pi}{6}\right)=-\sin\left(\frac{5\pi}{6}\right)=-\frac{1}{2}$

$\cos\left(-\frac{5\pi}{6}\right)=\cos\left(\frac{5\pi}{6}\right)=-\frac{\sqrt{3}}{2}$

$\tan\left(-\frac{5\pi}{6}\right)=-\tan\left(\frac{5\pi}{6}\right)=\frac{1}{\sqrt{3}}$

Notice a common pitfall: it’s easy to incorrectly change the sign of cosine for negative angles. Cosine is even, so it does not change.

Exam Focus

• Typical question patterns:
  • Evaluate $\sin(\theta)$, $\cos(\theta)$, $\tan(\theta)$ exactly for special angles (including angles outside $[0,2\pi]$).
  • Use relationships like $\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}$ to compute one trig value from others.
  • Identify domain restrictions for tangent and interpret vertical asymptotes.
• Common mistakes:
  • Forgetting tangent is undefined where $\cos(\theta)=0$.
  • Assuming tangent’s period is $2\pi$ instead of $\pi$.
  • Incorrectly applying even/odd symmetry (especially mixing up sine vs cosine).

Graphs of Sinusoidal Functions

From the unit circle to a wave graph

The unit circle definitions describe coordinates on a circle, but when you graph $y=\sin(x)$ or $y=\cos(x)$ you are doing something different: you’re using the angle input $x$ (typically in radians) and plotting the corresponding sine/cosine value as a vertical coordinate. As $x$ increases steadily, the point on the unit circle rotates steadily, and the $y$-coordinate (sine) rises and falls in a smooth repeating pattern.

This is why sine and cosine graphs look like “waves”: they are shadows (projections) of uniform circular motion.

Key features of $y=\sin(x)$ and $y=\cos(x)$

Both sine and cosine have:

• Midline at $y=0$
• Amplitude $1$ (distance from midline to max/min)
• Period $2\pi$
• Range from $-1$ to $1$

Where they differ is the starting point:

• Sine starts at $0$:

$\sin(0)=0$

• Cosine starts at $1$:

$\cos(0)=1$

A powerful way to think: cosine is a phase-shifted sine. In fact,

$\cos(x)=\sin\left(x+\frac{\pi}{2}\right)$

This is not just a neat identity—it explains why their graphs have the same shape but are horizontally shifted.

Intercepts, maxima/minima, and quarter-period reasoning

Sine and cosine are easiest to sketch by marking key points one quarter-period apart.

For $y=\sin(x)$ over one cycle from $0$ to $2\pi$:

• Zeros at $0, \pi, 2\pi$
• Maximum $1$ at $\frac{\pi}{2}$
• Minimum $-1$ at $\frac{3\pi}{2}$

For $y=\cos(x)$ over $0$ to $2\pi$:

• Maximum $1$ at $0$ and $2\pi$
• Zeros at $\frac{\pi}{2}$ and $\frac{3\pi}{2}$
• Minimum $-1$ at $\pi$

Many graphing mistakes happen when students try to “connect dots” without respecting the smoothness and symmetry. Sine and cosine are continuous and have a consistent wave shape—no sharp corners.

The tangent graph: repeating with asymptotes

The tangent function behaves differently:

$y=\tan(x)$

It has:

• Period $\pi$
• Zeros at integer multiples of $\pi$
• Vertical asymptotes where cosine is zero:

$x=\frac{\pi}{2}+k\pi$

for any integer $k$.

Between asymptotes, tangent increases from negative infinity to positive infinity. A common error is drawing tangent like a sine wave. It is not bounded; it has asymptotes and is unbounded.

Worked example: sketching a sinusoid using key features

Sketch one period of:

$y=2\sin(x)$

Start by interpreting the coefficient $2$ as amplitude. The midline stays at $y=0$, but the max/min become $2$ and $-2$.

Key points for $\sin(x)$ occur at $0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi$. Multiply the sine output by $2$:

• At $x=0$, $y=2\sin(0)=0$
• At $x=\frac{\pi}{2}$, $y=2\sin\left(\frac{\pi}{2}\right)=2$
• At $x=\pi$, $y=2\sin(\pi)=0$
• At $x=\frac{3\pi}{2}$, $y=2\sin\left(\frac{3\pi}{2}\right)=-2$
• At $x=2\pi$, $y=2\sin(2\pi)=0$

Then draw a smooth sinusoidal curve through these points.

Exam Focus

• Typical question patterns:
  • Identify amplitude, period, midline, and key points from a graph of sine/cosine.
  • Sketch a trig graph given a formula, often by plotting quarter-period points.
  • Describe key features of tangent graphs, including asymptotes and period.
• Common mistakes:
  • Treating tangent as bounded like sine/cosine (forgetting asymptotes).
  • Confusing amplitude with vertical shift (midline errors).
  • Using degree spacing on the horizontal axis while labeling with radians (unit mismatch).

Sinusoidal Function Transformations and Modeling

The general sinusoidal model and what each parameter does

A sinusoidal function is a transformed sine or cosine function used to model periodic behavior. A common form is:

$y=A\sin(B(x-C))+D$

or

$y=A\cos(B(x-C))+D$

Each parameter has a specific meaning:

• $A$: **amplitude** (with sign). The amplitude is $|A|$.
  • Controls vertical stretch/compression.
  • If $A<0$, the graph is reflected across its midline.
• $B$: controls the **period**. The period $P$ is:

$P=\frac{2\pi}{|B|}$

• $C$: **horizontal shift** (phase shift). The graph shifts right by $C$ if $C>0$.
• $D$: vertical shift. The midline is:

$y=D$

Why this matters: modeling is largely about interpreting these parameters in context. For example, in temperature data, $D$ is the average temperature, $|A|$ is the seasonal swing, and $P$ might be one year.

A common misconception is to think $A$ is the maximum value. It’s not—unless the midline is at zero. In general:

$\text{max}=D+|A|$

$\text{min}=D-|A|$

How to build a model from information (the “features first” approach)

When you’re given a real-world description or a graph and asked to write a sinusoidal equation, you usually don’t start by guessing $A,B,C,D$ randomly. Instead, extract features:

1) Midline: average of max and min.

$D=\frac{\text{max}+\text{min}}{2}$

2) Amplitude: half the distance between max and min.

$|A|=\frac{\text{max}-\text{min}}{2}$

3) Period: horizontal distance for one full cycle.

Then compute $B$ from:

$|B|=\frac{2\pi}{P}$

4) Phase shift: decide where the cycle starts. This is where choosing sine vs cosine can simplify the work:

• Cosine naturally starts at a maximum when $A>0$.
• Sine naturally starts at the midline increasing when $A>0$.

Students often make phase shift harder than it needs to be. If the data clearly starts at a peak, cosine is often the clean choice.

Interpreting transformations on graphs

Transformations are easier when you think in “layers.” Start with the parent function $\sin(x)$ or $\cos(x)$ and then apply changes:

• Multiply by $A$: vertical stretch and possibly reflection.
• Replace $x$ with $B(x-C)$: horizontal scale and shift.
• Add $D$: move up or down.

One subtle but important point: horizontal scaling works inversely. If you increase $|B|$, the period gets smaller because the input moves through a full trig cycle faster.

Worked example: create a sinusoidal model from max/min and period

Suppose the height $h$ (in meters) of a seat on a Ferris wheel varies sinusoidally with time $t$ (in seconds).

• Maximum height: $18$
• Minimum height: $2$
• One full rotation takes $40$ seconds
• At $t=0$, the seat is at its maximum height

Step 1: Midline

$D=\frac{18+2}{2}=10$

Step 2: Amplitude

$|A|=\frac{18-2}{2}=8$

Since the seat starts at a maximum at $t=0$, cosine with positive amplitude is convenient.

Step 3: Period and $B$

$P=40$

$B=\frac{2\pi}{40}=\frac{\pi}{20}$

Step 4: Phase shift
Starting at a maximum means no horizontal shift is needed for cosine: $C=0$.

So a model is:

$h(t)=8\cos\left(\frac{\pi}{20}t\right)+10$

Quick interpretation check:

• At $t=0$:

$h(0)=8\cos(0)+10=18$

• Half a rotation later at $t=20$:

$h(20)=8\cos(\pi)+10=2$

The model matches the given max and min.

Worked example: determine a model when the situation starts at the midline

A buoy bobs up and down. Its displacement from average water level (in centimeters) has amplitude $5$ cm and period $6$ seconds. At $t=0$ it is at the average level moving upward.

Starting at the midline moving upward matches the basic sine behavior, so choose a sine model with $D=0$.

Amplitude gives $A=5$.

Period $P=6$ gives:

$B=\frac{2\pi}{6}=\frac{\pi}{3}$

With no shift needed:

$d(t)=5\sin\left(\frac{\pi}{3}t\right)$

A common mistake here is using cosine and then forcing a phase shift when sine already fits the initial condition naturally.

Connecting models to graphs and data

On AP-style tasks, you might be given:

• A graph and asked to write an equation.
• An equation and asked to interpret parameters in context.
• A partial data set and asked to estimate a model.

In each case, keep anchoring your work to the meaning of the parameters:

• $D$ is the “average” or equilibrium level.
• $|A|$ is the size of the variation.
• $P$ is the cycle length, and $B$ encodes it.
• $C$ aligns the model with when a key feature occurs (peak, trough, or midline crossing).

Exam Focus

• Typical question patterns:
  • Given max/min and period (or a graph), write a sinusoidal model and justify parameter choices.
  • Interpret parameters $A,B,C,D$ in context (what they mean physically).
  • Compare sine vs cosine models for the same situation and explain phase shift.
• Common mistakes:
  • Computing period as $\frac{2\pi}{B}$ but forgetting absolute value, leading to sign confusion.
  • Treating $A$ as the maximum value instead of using $D+|A|$.
  • Getting phase shift direction wrong by mishandling $B(x-C)$ (the shift is tied to the entire factor inside parentheses).