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
• Radians, Arc Length & Angular Speed14 min · 3 objectivesConvert between degrees and radians and explain why radians are dimensionless · Compute arc length and sector area from a central angle in radians · Relate angular speed to linear speed on a rotating object
• Graphs of Sine & Cosine: Amplitude, Period, Phase14 min · 3 objectivesRead amplitude, period, midline and phase shift from an equation in the form a·sin(b(x − h)) + k · Compute the period as 2π/|b| and explain the reciprocal relationship · Write an equation for a sinusoid from its graph or from described features
• Tangent, Cotangent, Secant & Cosecant Graphs14 min · 3 objectivesLocate the vertical asymptotes of each of the four non-sinusoidal trigonometric graphs · State the period of tangent and cotangent and explain why it is π rather than 2π · Describe the range of secant and cosecant and why no values lie between −1 and 1
• Inverse Trigonometric Functions14 min · 3 objectivesState the restricted domains that make sine, cosine and tangent invertible · Give the exact range of each inverse trigonometric function · Evaluate compositions such as sin(arccos x) and explain when arcsin(sin x) ≠ x
• Solving Trigonometric Equations on an Interval14 min · 3 objectivesFind every solution of a trigonometric equation within a specified interval · Use reference angles and quadrant signs to generate all solutions from one · Handle equations where the argument is a multiple of the variable
• Sum, Difference & Double-Angle Formulas14 min · 3 objectivesApply the sum and difference formulas for sine and cosine · Derive the double-angle formulas from the sum formulas · Use these identities to find exact values and to simplify expressions
• Law of Sines & Law of Cosines14 min · 3 objectivesChoose between the Law of Sines and the Law of Cosines based on the given information · Recognize and resolve the ambiguous case of the Law of Sines · Compute the area of a triangle from two sides and the included angle
• Polar Graphs: Circles, Roses & Cardioids14 min · 3 objectivesIdentify the polar curve family from the form of its equation · Predict the number of petals of a rose curve from its coefficient · Explain what a negative value of r means and how it is plotted
• Rates of Change of Polar Functions14 min · 3 objectivesDetermine where a polar function r(θ) is increasing or decreasing · Explain what an increasing r means about the distance from the origin · Compute the average rate of change of r with respect to θ and interpret it
• Sinusoidal Regression & Model Fitting15 min · 3 objectivesFit a sinusoidal model to periodic data by estimating its four parameters · Judge whether periodic data is well described by a single sinusoid · Explain the limits of extrapolating a sinusoidal model

Formulas in Unit 3

Every term in Unit 3

All 87 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 at the origin; r = a cos θ or a sin θ is a circle through the origin; r = a ± b cos θ is a limaçon; r = a sin(nθ) is a rose (n petals if n is odd, 2n if 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.

Why a radian has no units

θ = s/r is a ratio of two lengths, so it is dimensionless. That is why s = rθ and A = ½r²θ need no conversion constant, while their degree versions require a factor of π/180.

Fraction-of-the-circle check

A 150° angle is 150/360 = 5/12 of the circle, so its arc is 5/12 of the circumference. Comparing this against s = rθ catches degree/radian confusion instantly.

Calculator angle mode

sin(30) is 0.5 in degree mode and −0.988 in radian mode. Neither is wrong; only one answers the question asked. Check the mode before every trigonometric computation.

Rolling without slipping

Distance traveled equals the arc length that unwinds: d = rθ. A wheel of radius 0.35 m turning 4.2 rad advances 1.47 m.

Angular speed on a rigid body

Every point shares the same ω because the object turns as one piece, but v = rω, so linear speed grows with distance from the axis. A propeller tip moves far faster than the hub.

Revolutions per minute to radians per second

Multiply by 2π/60. So 165 rpm is 165(2π)/60 ≈ 17.3 rad/s. Angular speed must be in radians per unit time before v = rω applies.

Reading amplitude from max and min

a = (max − min)/2, half the total range. Using the full range is the most common slip and doubles the amplitude.

Reading midline from max and min

k = (max + min)/2, the average of the extremes. This is the horizontal line the curve oscillates about, and it is the first parameter to extract.

Period from consecutive extremes

Consecutive maximum and minimum are HALF a period apart, not a full period. Getting this wrong doubles or halves b.

Why b and period are reciprocal

The argument bx reaches 2π when x = 2π/b, so a larger b compresses the graph and shortens the period. b = 4 makes the graph faster, with period π/2.

Factoring before reading a phase shift

sin(3x + π) must be rewritten as sin(3(x + π/3)) before the shift can be read. It is π/3 left, not π left — the horizontal compression scales the shift too.

Amplitude is never negative

For y = −4 sin x the amplitude is 4; the minus sign is a reflection across the midline, not a negative distance. Writing "amplitude = −4" loses a point.

Asymptotes of tangent

At x = π/2 + nπ, where cos x = 0. Since tan x = sin x/cos x, the asymptotes are exactly the zeros of the denominator — no memorization needed.

Asymptotes of cosecant

At x = nπ, where sin x = 0. Secant and tangent share the cosine zeros instead, at odd multiples of π/2.

Why tangent has period π

Advancing x by π flips the signs of both sine and cosine, and the two minus signs cancel in the quotient. Secant and cosecant get no such cancellation, so their period stays 2π.

Period of a transformed tangent

π/|b|, not 2π/|b|. So y = tan(3x) has period π/3. Applying the sinusoid rule to tangent is the single most common error in this topic.

Range of secant and cosecant

|y| ≥ 1, so no values lie strictly between −1 and 1. Since |sin x| ≤ 1, its reciprocal has absolute value at least 1.

Building a reciprocal graph from its sinusoid

Sketch the sinusoid first. Its zeros become asymptotes, its peaks at 1 become minima of the U at 1, and its troughs at −1 become maxima at −1.

Restricted domain of sine

[−π/2, π/2] — the stretch on which sine is one-to-one and still covers [−1, 1]. This restriction is what makes arcsin a function, and it becomes arcsin's range.

Restricted domain of cosine

[0, π]. So arccos returns values in quadrants I and II and can never return a negative angle — for a negative input it goes to quadrant II instead.

Range of arccos

[0, π]. arccos(−1/2) = 2π/3, not −π/3: the output is always between 0 and π, which is the asymmetry with arcsin that catches people out.

Range of arctan

(−π/2, π/2), open at both ends because tangent has asymptotes there. Domain is all real numbers, since tangent takes every value.

arcsin(sin x) is not always x

It equals x only when x is already in [−π/2, π/2]. arcsin(sin(5π/6)) = π/6, because the composition returns the equivalent angle inside the restricted range.

sin⁻¹ is not a reciprocal

sin⁻¹(x) means arcsin(x), the inverse function. 1/sin(x) is csc(x). The −1 in this notation is not an exponent.

Triangle method for a trig composition

For cos(arcsin(3/5)): let θ = arcsin(3/5), draw a right triangle with opposite 3 and hypotenuse 5, get adjacent 4, so cos θ = 4/5. The inverse function's range settles the sign.

Generating all solutions from a reference angle

Take the reference angle from the absolute value, decide which quadrants carry the right sign, produce both angles in one revolution, then add whole periods until you leave the interval.

Counting solutions on an interval

Count how many periods fit, then multiply by the solutions per period — two for an interior value, one for an extreme of the range. cos x = 0.4 on [0, 4π) has four.

Stretching the interval for sin(bx)

Substitute u = bx and note that u ranges over b times the original interval. Solve throughout that longer range, then divide every solution by b — otherwise you find only 1/b of them.

Quadratic in a trig function

2cos²x − cos x − 1 = 0 becomes 2u² − u − 1 = 0 under u = cos x. Factor, solve for u, then solve each trig equation separately.

Range check on a trig substitution

Discard any u outside [−1, 1] for sine or cosine — no angle produces it, so it contributes no solutions at all.

Endpoint values give one solution

sin x = 1 or −1 occurs once per period, at a maximum or minimum. Interior values occur twice. This changes the solution count and is easy to miss.

sin(A+B) is not sin A + sin B

Test it: sin(π/2 + π/2) = 0 while sin(π/2) + sin(π/2) = 2. Trigonometric functions are not linear, and the correct expansions mix sines with cosines.

Sign in the cosine sum formula

cos(A + B) = cos A cos B − sin A sin B — a MINUS for a plus. cos(A − B) uses addition. The reversal is the detail worth over-rehearsing.

Deriving the double-angle formulas

Set B = A in the sum formulas. sin 2A = 2 sin A cos A comes out immediately, and cos 2A = cos²A − sin²A follows the same way.

Three forms of cos 2A

cos²A − sin²A, 1 − 2sin²A, and 2cos²A − 1. All equivalent via the Pythagorean identity; picking the right one makes a simplification collapse in one step.

Choosing a cos 2A form

Match what you already have. If the expression contains sin²A, use 1 − 2sin²A. If it contains cos²A, use 2cos²A − 1.

Quadrant of a doubled angle

Does not follow from the quadrant of A. If sin A = 3/5 in quadrant II, then sin 2A = −24/25 and cos 2A = 7/25, putting 2A in quadrant IV.

Solving an equation with sin x and sin 2x

Replace sin 2x with 2 sin x cos x, move everything to one side, and factor: sin x(2cos x − 1) = 0. Never divide by sin x — that discards the solutions where it is zero.

Choosing between the two triangle laws

Law of Sines needs a complete side–angle pair, so use it for AAS, ASA and SSA. Law of Cosines needs none, so use it for SAS and SSS.

Law of cosines reduces to Pythagoras

With C = 90°, cos C = 0 and c² = a² + b² − 2ab·cos C becomes c² = a² + b². The Pythagorean theorem is the right-angle special case.

Solving the smaller angle first

After the Law of Cosines, use the Law of Sines on the SHORTER remaining side. Its angle must be acute, so no ambiguity about the supplement can arise.

No-triangle case in SSA

If the side opposite the given angle is shorter than the height a·sin A, the arc cannot reach the base and no triangle exists. SSA can give two triangles, one, or none.

Circle in polar form

r = a is centered at the origin with radius a. r = a cos θ or a sin θ is a circle through the origin, lying along the polar axis or perpendicular to it.

Diameter vs radius in r = a cos θ

The a is the DIAMETER. r = 6 sin θ is a circle of radius 3 centered at (0, 3), which follows from multiplying by r to get x² + y² = 6y.

Rose petal count

For r = a·cos(nθ): odd n gives n petals, even n gives 2n. So r = 5 sin(4θ) has 8 petals, each of length 5. Odd n retraces itself on the second half-revolution.

Cardioid vs limaçon

r = a ± b sin θ is a cardioid when a = b, with a single cusp at the origin. a < b gives an inner loop; a > b gives a dimple. The four quadrantal values reveal which.

Polar equation to rectangular

Multiply by r to create r² and r cos θ terms, then substitute r² = x² + y², x = r cos θ and y = r sin θ. This settles what an unfamiliar polar curve actually is.

Increasing r means receding

Where r = f(θ) is increasing, the curve moves away from the origin as the angle sweeps forward. Where f is decreasing, it moves toward the origin.

Distance from the origin is |r|

Not r. A decreasing negative r has increasing magnitude, so the curve is moving AWAY from the origin on the opposite side — decreasing r and approaching the origin are different claims.

Zeros of r

Where the curve passes through the origin. A sign change in r means the curve crosses to the opposite side, and for r = 1 + 2cos θ the negative stretch between the zeros is the inner loop.

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.
• Exact-value questions on inverse trig almost always use the special values 0, ±1/2, ±√2/2, ±√3/2, ±1 for sine and cosine, and 0, ±√3/3, ±1, ±√3 for tangent. Recognizing them removes the need for a calculator entirely.
• AP Precalculus asks about polar *rates of change* far more than about sketching polar curves. Expect to be given r = f(θ) and asked where the curve moves toward or away from the origin, with justification — and the justification must reference whether f is increasing or decreasing.

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