Trigonometric & Polar Functions

What this unit covers

The topics below follow the published Precalculus course framework for Unit 3. This unit is worth 30–35% of the exam, so budget your time against that rather than against how long the unit takes to teach.

Lessons in this unit

• The Unit Circle12 min · 3 objectivesInterpret cosine and sine as the x- and y-coordinates on the unit circle · Evaluate trig values at the common special angles · Locate the quadrantal-angle points on the unit circle
• Trigonometric Identities13 min · 3 objectivesState and apply the Pythagorean identity · Rewrite tangent and the reciprocal functions in terms of sine and cosine · Simplify a trigonometric expression using basic identities
• Sinusoidal Models14 min · 3 objectivesIdentify the amplitude, period, and midline of a sinusoidal function · Compute the period from the coefficient of x · Connect the parameters of y = A sin(Bx) + D to a real graph
• Polar Coordinates13 min · 3 objectivesConvert a point from polar form to rectangular form · Convert a point from rectangular form to polar form · Interpret r and θ as a distance and a direction

Formulas in Unit 3

Every term in Unit 3

All 37 terms we publish for Trigonometric & Polar Functions, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.

Pythagorean identity

sin²θ + cos²θ = 1, with tan²θ + 1 = sec²θ and 1 + cot²θ = csc²θ following by division.

Common polar graphs

r = a is a circle; r = a ± b cos θ gives limaçons and cardioids; r = a cos(nθ) gives roses with n petals if n is odd and 2n if n is even.

Radian measure

The angle subtending an arc equal to the radius. 2π radians is one full revolution, so 180° = π radians.

Arc length and sector area

s = rθ and A = ½r²θ, with θ in radians. These formulas are wrong in degrees, which is the usual source of error.

Unit circle definition

For an angle θ in standard position, the terminal point is (cos θ, sin θ). This extends the ratios to all angles, not just acute ones.

Reference angle

The acute angle to the x-axis. It gives the magnitude of the trig value; the quadrant supplies the sign.

Signs by quadrant

All positive in I; sine only in II; tangent only in III; cosine only in IV.

Special angle values

At 30°, 45° and 60° sine takes 1/2, √2/2 and √3/2 with cosine reversed. Worth memorizing rather than deriving under time pressure.

Sinusoidal general form

y = a·sin(b(x − h)) + k: |a| is amplitude, 2π/|b| is period, h is phase shift, k is midline.

Amplitude and midline

Amplitude is half the distance between maximum and minimum; the midline is their average, y = (max + min)/2.

Period of a trig function

2π/|b| for sine and cosine, π/|b| for tangent — tangent repeats twice as often, which is easy to miss.

Modeling with sinusoids

Use the midline for k, half the range for a, 2π divided by the observed period for b, and a known maximum or zero to solve for h.

Tangent function behavior

tan θ = sin θ/cos θ, so it has vertical asymptotes wherever cos θ = 0, at π/2 plus multiples of π, and period π.

Reciprocal functions

csc = 1/sin, sec = 1/cos, cot = 1/tan. Each has vertical asymptotes at the zeros of the function it inverts.

Inverse trig ranges

arcsin gives [−π/2, π/2], arccos gives [0, π], arctan gives (−π/2, π/2). The restriction is what makes the inverse a function.

Solving trig equations

Find every solution in one period using the reference angle and the correct quadrants, then add the period times any integer for the general solution.

Sum and difference identities

sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B. Note the sign flip in the cosine identity.

Double angle identities

sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ. The three forms of cos 2θ let you match whatever the problem gives you.

Law of sines

a/sin A = b/sin B = c/sin C. Used for AAS, ASA and SSA — and SSA is the ambiguous case that can give two triangles, one, or none.

Law of cosines

c² = a² + b² − 2ab·cos C. Used for SAS and SSS, and reduces to the Pythagorean theorem when C is a right angle.

Polar coordinates

A point as (r, θ): distance from the pole and angle from the polar axis. Unlike rectangular coordinates, the representation is not unique.

Polar to rectangular conversion

x = r cos θ and y = r sin θ; back the other way r² = x² + y² and tan θ = y/x, with the quadrant checked separately.

Rate of change in polar functions

Where r is increasing, the curve moves away from the origin as θ increases; where r is negative, the point plots opposite the given angle.

Coterminal angles

Angles differing by a whole number of full revolutions share a terminal ray and therefore all trig values.

Converting degrees and radians

Multiply by π/180 to go to radians, by 180/π to go back. Radians are required for arc length and calculus.

Angular and linear speed

Angular speed ω is radians per unit time; linear speed at radius r is v = rω. A point further out moves faster at the same angular speed.

Even-odd trig identities

cos(−θ) = cos θ (even); sin(−θ) = −sin θ and tan(−θ) = −tan θ (odd).

Cofunction identities

sin θ = cos(π/2 − θ) and similarly for the other pairs — the reason the "co" functions are named that way.

Why arcsin is restricted

Sine repeats, so without restricting the range to [−π/2, π/2] the inverse would assign many outputs to one input and not be a function.

Solving sin θ = k for all solutions

Find the reference angle, place it in both quadrants where sine has that sign, then add 2πn.

Trig identities as a proof tool

Work on one side only, converting everything to sine and cosine, until it matches the other. Do not move terms across the equals sign.

Sinusoidal regression vs algebraic fitting

Algebraic fitting uses the maximum, minimum and period to find each parameter; regression fits all points at once and is used when the data is noisy.

Ambiguous case of the law of sines

With SSA, the height h = b sin A determines the count: no triangle if a < h, one if a = h or a ≥ b, and two if h < a < b.

Area of a triangle from two sides and the included angle

A = ½ab sin C. Reduces to ½ base × height when C is a right angle.

Symmetry of polar graphs

Replacing θ with −θ tests symmetry about the polar axis; replacing θ with π − θ tests symmetry about the vertical line.

Negative r in polar coordinates

A negative r plots the point in the opposite direction from the given angle, which is how roses and limaçons trace their inner loops.

Finding where a polar curve crosses the origin

Solve r = 0 for θ. Those angles give the directions in which the curve passes through the pole.

What examiners penalize here

• Sign by quadrant: in QII cosine is negative and sine positive; in QIII both are negative; in QIV cosine is positive and sine negative. Combine a reference angle with the quadrant’s sign to get any exact value.
• When a proof stalls, convert every function to sine and cosine and look for the Pythagorean identity. Most AP simplification problems collapse once everything is written over a common sin/cos foundation.
• From a graph, read the midline and amplitude off the max and min: midline D = (max + min)/2 and amplitude A = (max − min)/2. Then find the period by measuring one full cycle and solve B = 2π/period.
• When converting rectangular → polar, arctan alone can give the wrong quadrant, since it only returns angles in (−π/2, π/2). Sketch the point first and add π when it lies in the second or third quadrant.

Practice Precalculus

Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.

Questions about this unit

All 4 units of AP Precalculus

1. Unit 1 · Polynomial & Rational Functions
2. Unit 2 · Exponential & Logarithmic Functions
3. Unit 3 · Trigonometric & Polar Functions
4. Unit 4 · Functions Involving Parameters, Vectors & Matrices