Unit 3: Trigonometric and Polar Functions

Periodic Phenomena and Key Graph Features

Periodic phenomena are occurrences or relationships that display a repetitive pattern over time or space. Typical examples include the undulating motion of waves, the circular rotation of clock hands, and the fluctuation in daylight hours throughout the year. In this unit, trigonometric functions become the main mathematical language for describing, graphing, and modeling that repetition.

A periodic function is a function that repeats a sequence of output values at fixed input intervals. The length of one full repetition is the period, which represents the length of one complete cycle.

When reading a periodic graph, you should be comfortable describing intervals where the function increases and decreases. Practically, that means identifying the set of input values where the graph moves from a local minimum to a local maximum (increasing), and from a local maximum to a local minimum (decreasing). You also want to notice concavity, which describes when the “rate” of increasing or decreasing is itself changing (the curve bends upward or downward). Even before calculus, concavity is visible from the shape of the graph.

Average rate of change is still the same idea as in earlier units: change in output divided by change in input. For a function $f$ over $[a,b]$:

$\text{Average rate of change} = \frac{f(b)-f(a)}{b-a}$

Exam Focus

• Typical question patterns:
  • Identify whether a situation or graph is periodic and estimate its period.
  • Describe intervals of increase/decrease and relate concavity to how the graph is bending.
  • Compute average rate of change over a time interval in a periodic context.
• Common mistakes:
  • Confusing “period” (cycle length) with “amplitude” (vertical size of oscillation).
  • Treating concavity as the same thing as increasing/decreasing.
  • Computing average rate of change with reversed subtraction order or mismatched interval endpoints.

Angles, Radians, and the Geometry of Rotation

Trigonometry starts with a simple idea: rotation. An angle tells you how much you turn from one ray to another. When describing an angle by rotation, the starting ray is the initial side and the ending ray is the terminal side. An angle is in standard position when its vertex is at the origin and its initial side lies along the positive $x$-axis. Counterclockwise rotation is considered positive, and clockwise rotation is considered negative.

Degrees vs. radians (and why radians are the “natural” unit)

A degree is defined by dividing a full circle into 360 equal parts. A radian is defined using circle geometry: an angle is 1 radian when it subtends an arc length equal to the radius.

For a circle of radius $r$, the circumference is:

$C = 2\pi r$

If a central angle has radian measure $\theta$, then the arc length it cuts off is:

$s = r\theta$

Equivalently, you can solve that for radian measure:

$\theta = \frac{s}{r}$

On the unit circle (where $r=1$), the radian measure equals the arc length. A complete turn corresponds to:

$2\pi \approx 6.28$

radians, which is the same as $360^\circ$.

Converting between degrees and radians

Because a full turn is both $360^\circ$ and $2\pi$ radians, half a turn gives:

$180^\circ = \pi$

So:

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

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

Radians matter for functions because when you graph sine and cosine, their natural period is $2\pi$ (not 360). Many later formulas also assume radians.

Coterminal angles and angle normalization

Angles that land at the same terminal side are coterminal. In degrees you add or subtract multiples of $360^\circ$; in radians you add or subtract multiples of $2\pi$. Angles can “spin more than once,” so measures can go beyond $2\pi$ radians.

If you want an equivalent angle in a standard interval (like between $0$ and $2\pi$), you repeatedly add/subtract $2\pi$ until you land in that interval.

Arc length and sector area

Radian measure lets you compute arc length and sector area efficiently.

$s = r\theta$

$A = \frac{1}{2}r^2\theta$

These formulas require $\theta$ in radians.

Worked example: arc length and sector area

A circle has radius $6$. Find the arc length and sector area for a central angle of $150^\circ$.

Convert to radians:

$150^\circ\cdot \frac{\pi}{180} = \frac{5\pi}{6}$

Arc length:

$s = 6\cdot \frac{5\pi}{6} = 5\pi$

Sector area:

$A = \frac{1}{2}\cdot 36\cdot \frac{5\pi}{6} = 15\pi$

Exam Focus

• Typical question patterns:
  • Convert between degrees and radians, then use $s=r\theta$ or $A=\frac{1}{2}r^2\theta$.
  • Find coterminal angles or an angle within $[0,2\pi)$.
  • Interpret what a radian “means” using arc length (especially on the unit circle).
• Common mistakes:
  • Using degree measures directly in $s=r\theta$ or $A=\frac{1}{2}r^2\theta$.
  • Dropping the radius factor and treating $s=\theta$ (only true on the unit circle).
  • Mixing up adding $\pi$ versus $2\pi$ when finding coterminal angles.

The Unit Circle and Defining Trigonometric Functions

Trigonometric functions become much more powerful when you define them using the unit circle rather than triangles. Triangle ratios work only for acute angles; the unit circle definition works for all real angles, including negative angles and angles greater than one full rotation.

The unit circle idea

The unit circle is the circle of radius 1 centered at the origin:

$x^2 + y^2 = 1$

For an angle $\theta$ in standard position, the terminal side intersects the unit circle at $(x,y)$. That point encodes the trig values:

$\cos(\theta) = x$

$\sin(\theta) = y$

Because every point on the unit circle satisfies $x^2+y^2=1$, you immediately get the Pythagorean identity:

$\cos^2(\theta)+\sin^2(\theta)=1$

Tangent and other trig functions

Once sine and cosine are defined, the other trig functions are built from them:

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

$\sec(\theta)=\frac{1}{\cos(\theta)}$

$\csc(\theta)=\frac{1}{\sin(\theta)}$

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

These definitions reveal domain restrictions:

• $\tan(\theta)$ and $\sec(\theta)$ are undefined when $\cos(\theta)=0$.
• $\csc(\theta)$ and $\cot(\theta)$ are undefined when $\sin(\theta)=0$.

Reference angles and quadrant signs

A reference angle is the acute angle between the terminal side of $\theta$ and the $x$-axis. It helps you use first-quadrant values even when an angle lies elsewhere. Even if an angle goes beyond one complete turn, its trig values match those of a coterminal angle, and reference angles let you quickly recover exact values.

Signs by quadrant:

• Quadrant I: sine positive, cosine positive
• Quadrant II: sine positive, cosine negative
• Quadrant III: sine negative, cosine negative
• Quadrant IV: sine negative, cosine positive

Special triangles and exact values

You are expected to know exact values for common angles (multiples of $\frac{\pi}{6}$ and $\frac{\pi}{4}$) using special triangles.

• 45-45-90 triangle: side ratio $1:1:\sqrt{2}$
• 30-60-90 triangle: side ratio $1:\sqrt{3}:2$

A practical method is: compute the legs in fraction form, then write the ordered pair on the unit circle corresponding to the radius endpoint.

Examples of key exact values:

$\cos\left(\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}$

$\sin\left(\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}$

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

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

Periodicity

Sine and cosine repeat every $2\pi$:

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

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

Tangent repeats every $\pi$:

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

Worked example: exact trig values using reference angles

Find $\sin\left(\frac{5\pi}{6}\right)$ and $\cos\left(\frac{5\pi}{6}\right)$.

Quadrant: $\frac{5\pi}{6}$ is in Quadrant II.

Reference angle:

$\alpha = \pi - \frac{5\pi}{6} = \frac{\pi}{6}$

Use known values at $\frac{\pi}{6}$ and apply Quadrant II signs:

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

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

Exam Focus

• Typical question patterns:
  • Use the unit circle to find exact values of sine, cosine, tangent, and reciprocals.
  • Convert a point on the unit circle to trig values (and vice versa).
  • Use reference angles and quadrant signs to evaluate trig expressions.
• Common mistakes:
  • Forgetting the sign based on quadrant.
  • Mixing up sine and cosine as $y$ versus $x$ on the unit circle.
  • Using tangent as $\frac{\cos}{\sin}$ instead of $\frac{\sin}{\cos}$.

Graphs and Key Features of Trigonometric Functions

In AP Precalculus, trigonometry is not just computing values; it is analyzing trig functions like any other function: domain, range, intercepts, symmetry, and key points.

Sine and cosine as functions

Using the unit circle, $\sin(\theta)$ and $\cos(\theta)$ take real-number inputs and output values between $-1$ and $1$.

Domain:

• All real numbers for sine and cosine.

Range:

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

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

Amplitude for the base graphs is 1, and the midline is:

$y=0$

Both have period:

$2\pi$

and frequency (cycles per unit of input) is:

$\text{frequency} = \frac{1}{\text{period}} = \frac{1}{2\pi}$

When graphing, it is common to mark the horizontal axis using key unit-circle angles, and keep the vertical axis scaled to $[-1,1]$.

Intercepts, maxima/minima, and symmetry

Cosine starts at a maximum because:

$\cos(0)=1$

Cosine is even:

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

Sine crosses the midline at the origin and initially increases:

$\sin(0)=0$

Sine is odd:

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

Phase shift relationship between sine and cosine

Sine and cosine have the same sinusoidal shape and can be viewed as horizontal shifts of one another:

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

and equivalently:

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

This idea is useful when choosing a model or rewriting an expression. Phase shift means a horizontal shift of the graph caused by adding/subtracting a constant inside the function.

Tangent as a function

Tangent is built from sine and cosine:

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

So it is undefined when $\cos(\theta)=0$:

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

Key features:

• Domain: all real $\theta$ except $\theta = \frac{\pi}{2}+k\pi$
• Range: all real numbers
• Period: $\pi$
• Intercepts: $\tan(0)=0$ and more generally zeros at:

$\theta = k\pi$

Tangent has vertical asymptotes where it is undefined.

Reciprocal functions (secant, cosecant, cotangent)

These appear often in function analysis and equation solving.

• Secant:

$\sec(\theta)=\frac{1}{\cos(\theta)}$

Domain excludes where cosine is zero:

$\theta \ne \frac{\pi}{2}+k\pi$

Range:

$\sec(\theta) \le -1 \text{ or } \sec(\theta) \ge 1$

Also:

$\sec(0)=1$

• Cosecant:

$\csc(\theta)=\frac{1}{\sin(\theta)}$

Domain excludes where sine is zero:

$\theta \ne k\pi$

Range:

$\csc(\theta) \le -1 \text{ or } \csc(\theta) \ge 1$

• Cotangent:

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

Domain excludes where sine is zero:

$\theta \ne k\pi$

Period:

$\pi$

Zeros occur when cosine is zero:

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

A common graphing strategy for reciprocal functions is: first locate where the original function (sine or cosine) is zero (those become vertical asymptotes), then locate where the original function reaches maxima/minima (those correspond to the “closest points” of the reciprocal branches).

Worked example: analyzing tangent

Find the domain restrictions and period of:

$f(\theta)=\tan(2\theta)$

Tangent’s base period is $\pi$. Replacing input by $2\theta$ compresses horizontally by factor 2, so the new period is:

$\text{period} = \frac{\pi}{2}$

Undefined where:

$\cos(2\theta)=0$

That happens when:

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

So:

$\theta = \frac{\pi}{4} + k\frac{\pi}{2}$

Exam Focus

• Typical question patterns:
  • Identify domain/range, period, symmetry, intercepts, and asymptotes from a trig function expression.
  • Use unit-circle structure to sketch sine/cosine and relate key angles to key points.
  • Analyze tangent, secant, and cosecant using where cosine/sine are zero.
• Common mistakes:
  • Treating tangent like it has period $2\pi$ instead of $\pi$.
  • Forgetting that transformations inside the function change the period.
  • Mixing up where reciprocal functions have asymptotes (where the original function is zero).

Trig Identities and Equivalent Representations

Identities let you rewrite expressions in more useful forms (for simplification, solving, or conversion in polar work).

Pythagorean identity

$\sin^2(\theta)+\cos^2(\theta)=1$

Sum identities

These are especially useful when rewriting expressions involving sums of angles:

$\sin(\alpha+\beta)=\sin(\alpha)\cos(\beta)+\cos(\alpha)\sin(\beta)$

$\cos(\alpha+\beta)=\cos(\alpha)\cos(\beta)-\sin(\alpha)\sin(\beta)$

Exam Focus

• Typical question patterns:
  • Use $\sin^2(\theta)+\cos^2(\theta)=1$ to rewrite an expression in terms of a single trig function.
  • Use sum identities to expand or recognize expressions with angle sums.
• Common mistakes:
  • Dropping parentheses when applying sum identities.
  • Confusing the sign structure in the cosine sum identity.

Transformations of Trig Graphs and Sinusoidal Parameters

A major goal of this unit is to understand how changing parameters changes a trig graph, especially sine and cosine, since that is the backbone of modeling periodic behavior.

The general sinusoidal form

A common transformed sine function is:

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

A common transformed cosine function is:

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

Interpretation:

• $A$ controls vertical stretch/compression and reflection across the $x$-axis if negative. Amplitude is $|A|$.
• $B$ controls horizontal stretch/compression. For sine/cosine, the period is:

$\text{period} = \frac{2\pi}{|B|}$

A negative $B$ reflects across the $y$-axis.

• $C$ is the phase shift (horizontal translation) when written in the factored form $B(x-C)$.
• $D$ is the vertical shift; the midline is $y=D$.

Building a sinusoid from features

Given a graph or context, it’s often easiest to use maximum and minimum values. If $y_{\max}$ is the maximum and $y_{\min}$ is the minimum, then:

$\text{amplitude} = \frac{y_{\max}-y_{\min}}{2}$

$\text{midline} = \frac{y_{\max}+y_{\min}}{2}$

The period is the horizontal distance for one full cycle. Frequency is the reciprocal of the period.

Choosing sine vs. cosine

Both can model the same periodic behavior; the difference is the starting position.

• Cosine naturally starts at a maximum when the inside input is 0.
• Sine naturally starts at the midline increasing when the inside input is 0.

Choosing the one that matches the initial condition with minimal phase shift usually leads to a simpler model.

Horizontal shifts: interpreting $B(x-C)$ correctly

A common algebra mistake is misreading phase shift when the inside isn’t factored.

Example:

$y = \sin(2x-\pi)$

Factor out 2:

$y = \sin\left(2\left(x-\frac{\pi}{2}\right)\right)$

So the shift is right by $\frac{\pi}{2}$.

Tangent transformations

Tangent can also be transformed, but it has different “base behavior” from sine and cosine. A common general form is:

$f(\theta)=a\tan(b(\theta-c))+d$

Interpretation:

• Vertical stretch/compression by $|a|$ and reflection across the $x$-axis if $a<0$.
• Tangent period changes to:

$\text{period} = \frac{\pi}{|b|}$

A negative $b$ reflects across the $y$-axis.

• Phase shift: horizontal translation by $c$.
• Vertical translation: shift by $d$; the midline becomes $y=d$.

Worked example: write a sinusoidal model from data

A Ferris wheel has radius 12 meters and its center is 15 meters above the ground. It completes one rotation every 20 seconds. A rider starts at the lowest point at time $t=0$. Write a height function $h(t)$.

Amplitude and midline:

$A = 12$

$D = 15$

Period $T=20$ gives:

$B = \frac{2\pi}{20} = \frac{\pi}{10}$

Starting at the lowest point suggests a reflected cosine:

$h(t)=15-12\cos\left(\frac{\pi}{10}t\right)$

Check at $t=0$:

$h(0)=15-12\cos(0)=3$

Exam Focus

• Typical question patterns:
  • Given a graph or scenario, write a sinusoidal function and interpret $A,B,C,D$.
  • Given an equation, identify amplitude, midline, period, and phase shift.
  • Compare two sinusoids or two tangents (which has larger amplitude? shorter period? shifted up?).
• Common mistakes:
  • Using $\frac{\pi}{B}$ as the period instead of $\frac{2\pi}{B}$ for sine/cosine.
  • Confusing amplitude with vertical shift.
  • Reading phase shift incorrectly when the inside is not factored.

Solving Trigonometric Equations and Interpreting Solutions

Solving trig equations relies on two core ideas: trig values repeat (periodicity), and most trig outputs correspond to multiple angles.

Solving basic equations using unit-circle values

Example:

$\sin(\theta)=\frac{1}{2}$

In one cycle, this occurs at:

$\theta = \frac{\pi}{6}$

$\theta = \frac{5\pi}{6}$

On $[0,2\pi)$, list those. For all real solutions:

$\theta = \frac{\pi}{6} + 2\pi k$

$\theta = \frac{5\pi}{6} + 2\pi k$

where $k$ is any integer.

Solving equations with transformed inputs

Example:

$\sin(2x)=\frac{1}{2}$

Solve for $2x$ first:

$2x = \frac{\pi}{6} + 2\pi k$

$2x = \frac{5\pi}{6} + 2\pi k$

Divide by 2:

$x = \frac{\pi}{12} + \pi k$

$x = \frac{5\pi}{12} + \pi k$

Using inverse trig to find principal values

Inverse trig functions undo trig functions but return principal values from a restricted range.

• $\arcsin(x)$ returns the unique angle $\theta$ in:

$\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$

such that $\sin(\theta)=x$.

• $\arccos(x)$ returns angles in:

$[0,\pi]$

• $\arctan(x)$ returns angles in:

$\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$

The reciprocal relationship holds when inputs are in the correct domains and ranges:

$f\left(f^{-1}(x)\right)=x$

and (with domain restriction for one-to-one behavior):

$f^{-1}\left(f(x)\right)=x$

Graphically, an inverse function is the reflection of the original function across:

$y=x$

The key practical point is that a calculator’s inverse trig output is usually only one solution (the principal value). You often must find additional solutions using symmetry and periodicity.

Solve using reciprocal functions

Sometimes an equation is easiest to rewrite using reciprocals:

$\csc(\theta)=\frac{1}{\sin(\theta)}$

$\sec(\theta)=\frac{1}{\cos(\theta)}$

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

This can expose when expressions are undefined (domain restrictions) and can simplify solving.

Worked example: solve on an interval

Solve on $[0,2\pi)$:

$2\cos(\theta)-1=0$

Isolate cosine:

$\cos(\theta)=\frac{1}{2}$

Solutions on one cycle:

$\theta=\frac{\pi}{3}$

$\theta=\frac{5\pi}{3}$

Worked example: solving with tangent and domain awareness

Solve for all real $x$:

$\tan(x)=\tan\left(\frac{\pi}{6}\right)$

Tangent repeats every $\pi$, so:

$x = \frac{\pi}{6} + \pi k$

Exam Focus

• Typical question patterns:
  • Solve $\sin(\theta)=a$ or $\cos(\theta)=a$ on $[0,2\pi)$ using the unit circle.
  • Solve transformed equations like $\sin(2x)=a$ and express solutions with integer parameters.
  • Use an inverse trig value, then decide whether a second (or infinite) set of solutions must be included.
  • Solve equations involving sec/csc/cot by rewriting in terms of sine and cosine.
• Common mistakes:
  • Giving only the principal value from inverse trig and forgetting the second angle in the cycle.
  • Using the wrong period when generating all solutions (especially for tangent).
  • Ignoring domain restrictions (including when rewriting with reciprocals).

Modeling Periodic Phenomena with Sinusoids

Sinusoidal models are a major real-world application of trig functions. The goal is to build a model, interpret its parameters, verify it against the situation, and then communicate conclusions clearly.

What makes a situation sinusoidal?

A situation is a good candidate when it repeats with a roughly constant cycle length (period), oscillates around a baseline (midline), and changes smoothly rather than abruptly. Common examples include tides, daylight hours, sound waves, alternating current, seasonal temperature patterns, and vertical motion from circular movement.

Extracting parameters from context

From max/min you get amplitude and midline. For example, if max temperature is 86 and min is 62:

$A = \frac{86-62}{2}=12$

$D = \frac{86+62}{2}=74$

If the period is 12 months:

$B = \frac{2\pi}{12} = \frac{\pi}{6}$

Interpreting phase shift in context

Phase shift is easiest if you align a real event with a base-function event.

• Cosine has a maximum at input 0.
• Sine crosses the midline going upward at input 0.

So if a maximum occurs at $x=x_0$, a cosine model often looks like:

$y = A\cos(B(x-x_0))+D$

If an upward midline crossing occurs at $x=x_0$, a sine model often looks like:

$y = A\sin(B(x-x_0))+D$

Worked example: temperature model with a given peak

The average monthly temperature varies from a minimum of 40 to a maximum of 80, repeating every 12 months. The maximum occurs at month 7 (July), where month 1 is January. Write a model $T(m)$.

Amplitude and midline:

$A = \frac{80-40}{2}=20$

$D = \frac{80+40}{2}=60$

Period and $B$:

$B = \frac{2\pi}{12} = \frac{\pi}{6}$

Put the maximum at month 7 with cosine:

$T(m)=20\cos\left(\frac{\pi}{6}(m-7)\right)+60$

Check:

$T(7)=20\cos(0)+60=80$

Using a model to answer questions

Once you have a model, you can answer questions such as when the function hits the midline, when it is above a threshold, or how it is changing over time. Many of these are solved by interpreting the graph or solving trig equations on a specified interval. You may also be asked to compute average rate of change over an interval or describe where the model is increasing/decreasing.

Interpreting, verifying, and reporting with models

A good modeling solution includes verification (plugging in key times to see if the model matches the situation) and reporting (stating relevant conclusions clearly for the intended audience without unnecessary algebraic detail or jargon).

Exam Focus

• Typical question patterns:
  • Build a sinusoidal function from max/min, period, and a key timing feature (peak or crossing).
  • Interpret parameters: explain what amplitude/period/vertical shift mean in context.
  • Verify a proposed model by checking key points (max, min, midline crossings).
  • Use the model to predict values or times within a specified interval.
• Common mistakes:
  • Using max as amplitude instead of half the max-minus-min distance.
  • Setting up phase shift without matching the event (peak vs. midline crossing).
  • Forgetting units: mixing months and years, seconds and minutes, radians and degrees.

Polar Coordinates: A New Way to Describe Location

Polar coordinates describe points by distance from the origin and direction, which often makes circular or rotational shapes easier to describe.

Polar coordinates and basic vocabulary

A point in polar form is $(r,\theta)$:

• $r$ is the radius (distance from the origin).
• $\theta$ is the angle measured from the positive $x$-axis (polar axis).

Polar representations are not unique:

$(r,\theta) \text{ and } (r,\theta+2\pi k) \text{ represent the same point}$

and

$(r,\theta) \text{ and } (-r,\theta+\pi) \text{ represent the same point}$

Negative $r$ means “go backward” along the ray.

Converting between polar and Cartesian

From polar to Cartesian:

$x = r\cos(\theta)$

$y = r\sin(\theta)$

From Cartesian to polar:

$r = \sqrt{x^2+y^2}$

For angle, a basic relationship is:

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

but you must choose $\theta$ in the correct quadrant based on the signs of $x$ and $y$.

Notation reference (common forms)

Worked example: convert polar to Cartesian

Convert $\left(4,\frac{2\pi}{3}\right)$ to Cartesian.

$x = 4\cos\left(\frac{2\pi}{3}\right)=-2$

$y = 4\sin\left(\frac{2\pi}{3}\right)=2\sqrt{3}$

So the point is $(-2,2\sqrt{3})$.

Worked example: convert Cartesian to polar

Convert $(-3,3)$ to polar.

Radius:

$r = \sqrt{(-3)^2+3^2} = \sqrt{18}=3\sqrt{2}$

Angle:

$\tan(\theta)=\frac{3}{-3}=-1$

Reference angle is $\frac{\pi}{4}$. The point is in Quadrant II, so:

$\theta = \pi-\frac{\pi}{4} = \frac{3\pi}{4}$

One polar form is:

$\left(3\sqrt{2},\frac{3\pi}{4}\right)$

Polar form and complex numbers

Complex numbers can be written in rectangular form $a+bi$ and also in a polar-style trigonometric form that uses angle and magnitude:

$r(\cos(\theta)+i\sin(\theta))$

This relies on the same conversion ideas, with $\cos(\theta)$ and $\sin(\theta)$ representing horizontal and vertical components.

Exam Focus

• Typical question patterns:
  • Convert points between polar and Cartesian.
  • Recognize equivalent polar coordinates using $2\pi$ shifts or negative $r$.
  • Use polar conversion ideas in contexts involving vectors or complex-number magnitude/angle form.
• Common mistakes:
  • Forgetting quadrant when finding $\theta$ from $\tan(\theta)=\frac{y}{x}$.
  • Treating $r$ as always positive.
  • Mixing up $x=r\cos(\theta)$ and $y=r\sin(\theta)$.

Polar Equations and Graphs: Circles, Lines, Roses, and Limacons

A polar equation describes a relationship between $r$ and $\theta$. To graph $r=f(\theta)$, you typically choose key angles, compute $r$, plot points, and then use symmetry to reduce work.

How to sketch a polar graph (practical process)

1. Choose key angles (often multiples of $\frac{\pi}{6}$ or $\frac{\pi}{4}$).
2. Compute $r$ for those angles.
3. Plot each point by moving out $r$ units in direction $\theta$.
4. Look for symmetry.

Symmetry tests (useful, but interpret carefully)

• Symmetry about the polar axis often occurs if replacing $\theta$ with $-\theta$ leaves the equation unchanged.
• Symmetry about the line $\theta = \frac{\pi}{2}$ often occurs if replacing $\theta$ with $\pi-\theta$ leaves the equation unchanged.
• Symmetry about the origin often occurs if replacing $r$ with $-r$ (or replacing $\theta$ with $\theta+\pi$) leaves the equation unchanged.

Key basic polar graphs

1) Circles centered at the origin: $r=a$
If $r$ is constant, every point is the same distance from the origin, forming a circle of radius $|a|$.

2) Lines through the origin: $\theta = \theta_0$
If $\theta$ is constant, you get a line through the origin at angle $\theta_0$.

3) Circles through the origin: $r=a\cos(\theta)$ and $r=a\sin(\theta)$
These produce circles not centered at the origin. Converting reveals the circle’s center and radius.

For $r=a\cos(\theta)$:

$r^2 = ar\cos(\theta)$

Using $r^2=x^2+y^2$ and $r\cos(\theta)=x$:

$x^2+y^2=ax$

Complete the square:

$\left(x-\frac{a}{2}\right)^2+y^2=\left(\frac{a}{2}\right)^2$

This is a circle centered at $\left(\frac{a}{2},0\right)$ with radius $\frac{|a|}{2}$. If $a>0$, it sits to the right of the origin; if $a<0$, it sits to the left.

Similarly, $r=a\sin(\theta)$ converts to:

$x^2+\left(y-\frac{a}{2}\right)^2=\left(\frac{a}{2}\right)^2$

This circle is centered at $\left(0,\frac{a}{2}\right)$, above the origin if $a>0$ and below if $a<0$.

Rose curves

Rose curves have the form:

$r = a\cos(n\theta)$

or

$r = a\sin(n\theta)$

Petal count:

• If $n$ is odd, the curve has $n$ petals.
• If $n$ is even, the curve has $2n$ petals.

The maximum radius (petal length) is $|a|$.

Angle placement of petal tips depends on parity:

• If $n$ is odd, tips occur at angles spaced by:

$\frac{2\pi}{n}$

• If $n$ is even, tips occur at angles spaced by:

$\frac{\pi}{n}$

Roses also exhibit symmetry; which axis/line symmetry appears depends on whether the equation uses sine or cosine and on the value of $n$.

Limacons

Limacons have the form:

$r=a\pm b\cos(\theta)$

or

$r=a\pm b\sin(\theta)$

Shape classification (using magnitudes):

• Dent (no loop) if $|a|>|b|$
• Cardioid (heart shape) if $|a|=|b|$
• Loop if $|a|<|b|$

Radius extremes:

$r_{\max}=|a|+|b|$

$r_{\min}=|a|-|b|$

Worked example: identify a polar curve by conversion

Identify the Cartesian form and describe the graph:

$r = 4\cos(\theta)$

Multiply by $r$:

$r^2=4r\cos(\theta)$

Convert:

$x^2+y^2=4x$

Complete the square:

$\left(x-2\right)^2+y^2=4$

It’s a circle centered at $(2,0)$ with radius $2$.

Worked example: sketch a rose curve conceptually

Sketch:

$r = 2\sin(3\theta)$

Here $n=3$ (odd), so there are 3 petals, each with maximum radius 2. Petal tips occur when $\sin(3\theta)=1$:

$3\theta = \frac{\pi}{2} + 2\pi k$

So:

$\theta = \frac{\pi}{6} + \frac{2\pi}{3}k$

Exam Focus

• Typical question patterns:
  • Recognize and sketch graphs of $r=a$, $\theta=\theta_0$, $r=a\cos(\theta)$, $r=a\sin(\theta)$.
  • Convert between polar and Cartesian to identify a curve.
  • Determine petal count and max radius for rose curves and classify limacons.
  • Use symmetry to sketch efficiently, but still validate with a few computed points.
• Common mistakes:
  • Forgetting to multiply by $r$ when converting equations like $r=a\cos(\theta)$.
  • Miscounting petals for even $n$.
  • Ignoring negative $r$ values when plotting, which can hide parts of the graph.
  • Misclassifying limacons by comparing $a$ and $b$ without using absolute values.

Rates of Change in Polar Functions

Polar functions can be analyzed using many of the same ideas as other functions: increasing/decreasing behavior, positive/negative intervals, extrema, and average rate of change.

A polar function has the form $r=f(\theta)$, where the input is an angle and the output is a radius (distance from the origin, possibly negative). Relative extrema in polar contexts often correspond to points that are closest to or farthest from the origin, and they occur when the function transitions from increasing to decreasing or from decreasing to increasing.

Average rate of change of $r$ with respect to $\theta$ over $[\theta_1,\theta_2]$ is:

$\frac{\Delta r}{\Delta \theta} = \frac{f(\theta_2)-f(\theta_1)}{\theta_2-\theta_1}$

Interpreting sign and change (idea summary):

• If $r$ is positive, the point lies in the direction of $\theta$.
• If $r$ is negative, the point lies in the direction of $\theta+\pi$.
• If $r$ is increasing as $\theta$ increases, the distance from the origin is increasing.
• If $r$ is decreasing as $\theta$ increases, the distance from the origin is decreasing.

Exam Focus

• Typical question patterns:
  • Compute and interpret average rate of change of a polar radius over an interval of angles.
  • Identify where a polar function is increasing/decreasing and connect that to moving away from or toward the origin.
  • Use relative extrema of $r=f(\theta)$ to reason about nearest/farthest points.
• Common mistakes:
  • Treating negative $r$ as “invalid” instead of recognizing it flips direction.
  • Mixing up “increasing $r$” with “counterclockwise rotation” (they describe different aspects of motion).
  • Computing average rate of change with incorrect endpoints or a missing denominator.

Connecting Polar and Trigonometric Thinking

Trigonometry and polar coordinates are two views of the same geometry.

• Trigonometry connects angles to coordinates on a circle.
• Polar coordinates describe points using distance and angle.

The bridge formulas are:

$x=r\cos(\theta)$

$y=r\sin(\theta)$

These are essentially unit-circle trig scaled by $r$.

Interpreting projections

You can interpret these conversions as vector components:

• $r\cos(\theta)$ is the horizontal component of a length-$r$ vector at angle $\theta$.
• $r\sin(\theta)$ is the vertical component.

Using trig identities in polar contexts

Because polar equations often involve sine and cosine, identities help you simplify and convert. The most essential identity here is:

$\sin^2(\theta)+\cos^2(\theta)=1$

Also remember the standard substitutions:

$r^2=x^2+y^2$

$\cos(\theta)=\frac{x}{r}$

$\sin(\theta)=\frac{y}{r}$

Worked example: convert a mixed polar equation

Convert to Cartesian and describe the curve:

$r^2 = 9\cos^2(\theta)$

Use $r^2=x^2+y^2$ and $\cos(\theta)=\frac{x}{r}$:

$x^2+y^2 = 9\left(\frac{x}{r}\right)^2$

Eliminate $r$ by replacing $r^2$ with $x^2+y^2$:

$x^2+y^2 = 9\frac{x^2}{x^2+y^2}$

Multiply both sides by $x^2+y^2$:

$(x^2+y^2)^2 = 9x^2$

What can go wrong in conversions

Conversions are algebra-heavy, so small slips cause big errors:

• Forgetting that $r^2=x^2+y^2$ (not $r=x^2+y^2$).
• Substituting $\cos(\theta)=\frac{x}{r}$ or $\sin(\theta)=\frac{y}{r}$ but not eliminating $r$.
• Dividing by an expression that might be zero (for example, dividing by $r$ without considering $r=0$).

Exam Focus

• Typical question patterns:
  • Explain how unit-circle trig leads to the polar-to-Cartesian conversion formulas.
  • Convert polar equations to Cartesian (especially circles and other recognizable forms).
  • Interpret $r\cos(\theta)$ and $r\sin(\theta)$ as components in context.
• Common mistakes:
  • Using the wrong substitution (mixing up $r\cos(\theta)=x$ and $r\sin(\theta)=y$).
  • Losing solutions when multiplying/dividing by expressions involving $r$.
  • Treating a polar curve like a standard $y=f(x)$ graph and expecting a “vertical line test” perspective.