Unit 3 · Large assessed unit

Trigonometric and Polar Functions

The unit that introduces periodicity. It builds the unit circle, turns it into sinusoidal functions with four adjustable parameters, uses those to model repeating phenomena, and then changes coordinate systems entirely to look at polar functions.

What a strong answer looks like

A strong Unit 3 answer identifies the quadrant or the period before producing a value, and interprets amplitude and midline in the language of the situation being modelled.

Practice this unit Course overview

Topics in this unit

Periodic Phenomena

Know

A periodic function repeats over a fixed interval. Midline is the average of the extremes, amplitude is half their difference, and period is the input length of one full cycle.

Apply

From a described situation, extract midline and amplitude from the maximum and minimum, and the period from how often the pattern repeats.

Watch out

Using the full difference between maximum and minimum as the amplitude doubles it. Amplitude is a distance from the midline, not across the whole range.

Study move

Given a tide varying between 3 and 11 metres every 12 hours, state midline, amplitude, and period before writing any equation.

The Unit Circle

Know

On the unit circle the coordinates of the terminal point are the cosine and sine of the angle. Symmetry generates every quadrant from the first, so structure replaces memorisation.

Apply

Find the reference angle, take the magnitude from the special triangles, then attach the sign the quadrant requires.

Watch out

Sign errors dominate this material. Deciding the quadrant first eliminates nearly all of them.

Study move

For four angles spread across the quadrants, state the quadrant, the reference angle, and the signs of sine and cosine before evaluating.

Sinusoidal Parameters

Know

In a sinusoid the coefficient in front sets amplitude, the coefficient of the variable sets period through the relation that period equals a full turn divided by it, and the constant at the end sets the midline.

Apply

Read amplitude and midline directly, and compute the period from the inside coefficient rather than reading it off.

Watch out

Treating the inside coefficient as the period, rather than dividing into a full turn, inverts the relationship.

Study move

For a sinusoid with an inside coefficient of four, compute the period and confirm it against a graph.

Phase Shift

Know

A phase shift is only visible once the inside coefficient has been factored out of the argument. Before factoring, the constant in the argument is not the shift.

Apply

Rewrite the argument in factored form, then read the shift as the value subtracted from the variable.

Watch out

Reading the shift straight from the unfactored constant is the most common Unit 3 error, and every distractor set exploits it.

Study move

Rewrite an argument of the form four x minus pi in factored form and state the actual shift.

Modeling with Sinusoids

Know

Building a model means choosing sine or cosine and the sign in front so that the starting position matches the situation, then setting the four parameters.

Apply

Decide first where the phenomenon starts: at a maximum, a minimum, or the midline. That choice determines the function and its sign before any parameter is set.

Watch out

A model with correct amplitude, period, and midline can still be wrong if it starts at the top when the situation starts at the bottom.

Study move

Model a Ferris wheel where the rider boards at the lowest point, and check the model returns the boarding height at time zero.

Trigonometric Equations and Identities

Know

Trigonometric equations have infinitely many solutions because the functions repeat, so the interval stated in the problem determines which are reported. Identities rewrite an expression into a solvable form.

Apply

Solve for the trigonometric expression first, then find every angle in the stated interval that produces it.

Watch out

A calculator returns one principal value. The second solution in the interval must be found from symmetry, not from the calculator.

Study move

Solve a quadratic in sine, then list every solution in one full turn rather than stopping at the first.

Polar Functions

Know

A polar function gives a radius as a function of angle. Whether the radius is increasing or decreasing over an interval describes whether the curve moves away from or toward the pole.

Apply

Convert between systems using the relations that x is the radius times cosine and y is the radius times sine, and that the radius squared is the sum of the squares of the coordinates.

Watch out

A single point has infinitely many polar representations, and a negative radius reflects the point through the pole rather than being invalid.

Study move

Convert a polar equation such as a constant times cosine into rectangular form and identify the resulting circle.

Emphasized in this unit

Connections and techniques that receive extra attention in this AP unit.

• Reading midline and amplitude from a described situation before writing any equation
• Producing exact values from reference angle and quadrant rather than from a memorised table
• Factoring the inside coefficient out before reading a phase shift
• Choosing sine or cosine, and its sign, from where the phenomenon starts

Varies by course

Related topics some schools attach to this unit and others leave out. Covered on request rather than assumed.

Reciprocal function graphs. Secant, cosecant, and cotangent graphs are covered in depth in some courses and only named in others.
Proof of identities. Some courses require formal verification; the framework emphasises using identities over proving them.
Polar graph families. Roses, cardioids, and limacons are classified in detail in some courses and treated lightly in others.

Mastery checklist

• Extract midline, amplitude, and period from a described periodic situation.
• Give exact sine and cosine values for the special angles in every quadrant.
• State amplitude, period, phase shift, and midline from a sinusoidal equation.
• Build a sinusoidal model that starts in the right place for the situation.
• Report every solution of a trigonometric equation within a stated interval.
• Convert between polar and rectangular coordinates and equations.
• Describe how a curve moves as a polar radius increases or decreases.

Check yourself

• Why is amplitude half the difference between maximum and minimum rather than the whole difference?
• Why must the inside coefficient be factored out before the phase shift is read?
• Why does a calculator give only one solution to a trigonometric equation?
• What does a negative radius mean in polar coordinates?

Modeling drill

A Ferris wheel of radius 20 metres has its centre 25 metres above the ground and turns once every 40 seconds, with a rider boarding at the lowest point. Build the height model, then check it returns the boarding height at time zero.

PeriodMidlineAmplitudePhase shiftReference angleUnit circlePolar coordinatesPole

Unit 2: Exponential and Logarithmic Functions

Unit 4: Functions Involving Parameters, Vectors, and Matrices