Precalculus

295 core topics + 72 prerequisite topics taught as needed · approximately 109 hours of instruction including spaced review

How the course runs

An adaptive diagnostic (up to 40 questions) places the student on the course's knowledge graph - topics already known are credited, and instruction begins exactly at the learning frontier. Every topic is taught with a worked-example lesson and auto-graded practice; a topic is mastered at 75%+ and then maintained through spaced reviews on an expanding schedule. Mixed checks follow every 6 lessons; each unit ends with a 12-item quiz, and course-wide assessments appear at 25%, 50%, 75%, and 100% mastery. Prerequisite gaps below the course are detected and taught rather than skipped, so completion certifies the whole tower, not just the top.

Core curriculum

Trigonometry · 15 topics

Right-Triangle Trigonometry [M]	SOH-CAH-TOA: the three trig ratios of an acute angle in a right triangle.
Solving for Sides with Trig [M]	Using a known angle and one side to find another side.
Degrees & Radians [E]	Two ways to measure the same angle: 180° equals π radians.
The Unit Circle [M]	Exact sine, cosine, and tangent values at the special angles.
Trig of Any Angle [H]	Reference angles plus quadrant signs extend trig beyond 90°.
Graphs of Sine & Cosine [M]	Reading amplitude, period, and midline from y = a sin(bx) + c.
Phase Shifts & Other Trig Graphs [M]	Horizontal (phase) shifts of trig graphs, and the period of tangent.
The Pythagorean Identity [M]	sin²θ + cos²θ = 1 links sine and cosine of the same angle.
Basic Trig Identities [M]	Quotient, reciprocal, and even-odd identities.
Sum & Difference Formulas [H]	Expanding sin(A ± B) and cos(A ± B) to reach non-special angles.
Double-Angle Formulas [H]	sin 2x = 2 sin x cos x and cos 2x = 1 − 2 sin²x.
Trig Equations [H]	Isolating a trig function and reading solutions off the unit circle.
Law of Sines [H]	In any triangle, each side over the sine of its opposite angle is constant.
Law of Cosines [H]	c² = a² + b² − 2ab cos C generalizes the Pythagorean theorem.
Inverse Trig Functions [M]	arcsin, arccos, and arctan undo the trig functions on restricted ranges.

Precalculus · 12 topics

Asymptotes of Rational Functions [M]	The vertical asymptotes and the end behavior (horizontal or slant asymptote) of a rational function.
Graphs of Rational Functions [M]	Holes, asymptotes, and intercepts tell the whole story of the graph.
Polynomial Inequalities [M]	Sign charts: zeros split the number line into test intervals.
Vectors: Components & Magnitude [M]	A vector is a displacement: components ⟨Δx, Δy⟩ and a length.
Vector Operations [M]	Scaling, adding, and dotting vectors - all component by component.
Parametric Equations [M]	Describing a moving point by giving x and y as functions of time.
Polar Coordinates [H]	Locating points by distance from the origin and angle from the x-axis.
Polar Graphs [M]	Recognizing circles, lines, and rose curves from polar equations.
Circles & Ellipses [M]	Reading centers, radii, and intercepts from conic equations.
The Binomial Theorem [H]	Expanding (x + a)ⁿ without multiplying it out term by term.
Sigma Notation & Series [M]	Σ compresses a sum: read the limits, add up the terms.
Average Rate of Change [M]	The slope of the secant line: (f(b) − f(a)) / (b − a).

Precalculus: Vectors & Matrices · 13 topics

Vector Components from Two Points [E]	Head minus tail, coordinate by coordinate.
Magnitude of a Vector [E]	A vector's length comes straight from the Pythagorean theorem.
Adding & Subtracting Vectors [E]	Vectors combine component by component - tip-to-tail in coordinates.
Scalar Multiples & Combinations [E]	A scalar stretches every component; combinations mix scaled vectors.
The Dot Product [M]	Multiply matching components and add - two vectors in, one number out.
Perpendicular Vectors [M]	Two vectors are perpendicular exactly when their dot product is zero.
Classifying the Angle Between Vectors [M]	The sign of the dot product tells acute, right, or obtuse.
2×2 Matrices: Addition & Scalar Multiples [E]	Same-shape matrices add entry by entry; a scalar hits every entry.
Multiplying a Matrix by a Vector [M]	Each output component is a row of the matrix dotted with the vector.
2×2 Matrix Multiplication [M]	Row of the left matrix times column of the right, entry by entry.
The 2×2 Determinant [M]	Down-diagonal product minus up-diagonal product: ad − bc.
Determinant as Parallelogram Area [M]	The parallelogram on ⟨a, b⟩ and ⟨c, d⟩ has area |ad − bc|.
2×2 Systems as Matrix Equations [H]	A pair of linear equations is one matrix equation with one solution.

Precalculus: Conic Sections · 12 topics

Circle Equations: Center & Radius [E]	Read the center and radius straight off (x − h)² + (y − k)² = r².
Circle Through a Given Point [M]	The radius is the distance from the center to any point on the circle.
Circles by Completing the Square [M]	Turn x² + y² + Dx + Ey + F = 0 back into center-radius form.
Parabolas: Focus & Directrix [M]	In x² = 4py the focus sits p above the vertex and the directrix p below.
Parabolas with a Shifted Vertex [M]	Vertex, focus, and directrix stay p apart no matter where the vertex sits.
Ellipses in Standard Form [E]	The larger denominator points along the major axis: a² under it, b² under the other.
Foci of an Ellipse: c² = a² − b² [M]	The foci sit inside the ellipse on the major axis, c² = a² − b² from center.
Hyperbolas in Standard Form [E]	The positive squared term tells you the axis the two branches open along.
Foci of a Hyperbola: c² = a² + b² [M]	Hyperbola foci sit beyond the vertices: c² adds a² and b².
Asymptotes of a Hyperbola [M]	The branches hug the lines y = ±(b/a)x through the center.
Eccentricity as an Exact Fraction [M]	e = c/a measures shape: below 1 for ellipses, above 1 for hyperbolas.
Classifying a Conic from Its Equation [M]	Compare the squared terms: their signs and coefficients name the conic.

Precalculus: Polar Coordinates & Complex Numbers · 10 topics

Adding & Subtracting Complex Numbers [E]	Combine real parts and imaginary parts separately.
Multiplying Complex Numbers [M]	FOIL, then replace i² with −1.
Powers of i [E]	The powers of i repeat every four steps.
Modulus of a Complex Number [M]	The distance from the origin: √(a² + b²).
Complex Conjugates [M]	Flip the sign of i; the product is real.
Dividing Complex Numbers [M]	Multiply top and bottom by the denominator's conjugate.
Polar to Rectangular [M]	x = r cos θ, y = r sin θ.
Exact Radical Coordinates [M]	The 30° and 60° coordinates carry a √3.
Rectangular to Polar (r) [M]	The radius is the distance to the origin.
Reference Angles [M]	The acute angle to the nearest x-axis.

Precalculus: Trig Identities & Equations in Action · 12 topics

Recovering tan θ from One Ratio [M]	Combine the Pythagorean identity with tan θ = sin θ / cos θ, then let the quadrant fix the sign.
Secant & Cosecant from One Ratio [M]	Find the missing ratio by the Pythagorean identity, then flip it: sec = 1/cos, csc = 1/sin.
Cotangent from One Ratio [M]	cot θ = cos θ / sin θ - the quotient identity read the other way up.
Exact Values by Decomposition [H]	Split an unusual angle into a sum of special angles, then expand.
Combining Two Known Angles [H]	Given sines of two angles, build sin(A ± B) and cos(A ± B) as exact fractions.
Computing sin(2x) [M]	sin 2x = 2 sin x cos x - recover the missing factor with its correct sign first.
Computing cos(2x) [M]	cos 2x = 1 − 2 sin²x = 2 cos²x − 1 - one squared ratio is enough.
Identities That Collapse to a Number [M]	sec²−tan² = 1, csc²−cot² = 1, and each function times its reciprocal is 1.
Simplifying to One Function [M]	Rewrite a product or quotient in terms of sine and cosine, then cancel.
Counting Solutions on [0°, 360°) [M]	Each attainable value of sine or cosine is hit twice per turn - except at the peaks.
The Smallest Solution in Degrees [M]	Isolate the function, find the reference angle, then take the least angle in range.
Verifying the Right Formula [M]	Spot the correct expansion and reject the near-miss sign and swap errors.

Precalculus: Exponential & Logarithmic Functions · 13 topics

Evaluating Exponential Functions [E]	Plug integer inputs into f(x) = a·bˣ - including 0 and negatives.
Graphs: y-Intercept & Asymptote [M]	y-intercept a, growth when b > 1, decay when 0 < b < 1, floor at y = 0.
Evaluating Logarithms [M]	log_b(x) asks: to what power must b be raised to get x?
Log Laws in Computation [M]	Logs turn products into sums, quotients into differences, powers into multiples.
Exponentials & Logs as Inverses [M]	log_b and b^x undo each other - reflections across the line y = x.
Exponential Equations: Same Base [M]	Match the bases, then set the exponents equal.
Exponential Equations with Logs [M]	Take a logarithm of both sides to bring the exponent down.
The Natural Base e and ln [M]	e ≈ 2.718 is the natural base; ln is log base e, its exact inverse.
Exponential Growth & Decay Models [M]	Model y = a·bᵗ: multiply the start by the factor once per time step.
Continuous Compound Interest [M]	Compounding at every instant uses the natural base: A = P·e^(rt).
Doubling Time [M]	Doubling every T means y = a·2^(t/T) - count the doublings first.
Half-Life [M]	Half-life is the time to halve once - y = a·(1/2)^(t/H).
Solving Logarithmic Equations [M]	Rewrite log_b(expr) = k as expr = b^k, then solve.

Precalculus: Rational Functions, Series & the Binomial Theorem · 13 topics

Vertical Asymptotes vs. Holes [M]	A canceling factor makes a hole; a surviving denominator factor makes an asymptote.
Horizontal Asymptotes by Degree [M]	Compare the top and bottom degrees to read off the horizontal asymptote.
Intercepts of Rational Functions [M]	x-intercepts come from the numerator's zeros; the y-intercept is f(0).
The Coordinates of a Hole [M]	Cancel the common factor, then plug the x-value into what remains.
Combinations C(n, r) [M]	Count unordered selections with C(n, r) = n! / (r!(n − r)!).
Permutations P(n, r) [M]	Count ordered arrangements with P(n, r) = n! / (n − r)!.
Permutation or Combination? [M]	Decide whether order matters, then pick P(n, r) or C(n, r).
Pascal's Triangle Entries [M]	Every Pascal entry is a C(n, k), and each row sums to 2ⁿ.
Extracting a Binomial Coefficient [H]	One term of (x ± a)ⁿ: C(n, k)·aⁿ⁻ᵏ, with the sign tracked.
Finite Arithmetic Series [M]	Sum an arithmetic series with S = n(first + last)/2.
Finite Geometric Series [M]	Sum n geometric terms with S = a(1 − rⁿ)/(1 − r).
Evaluating Sigma Notation [M]	Read the limits, apply the standard sum formulas, and add.
Infinite Geometric Series [M]	When |r| < 1 the endless sum converges to a/(1 − r).

Precalculus: Trigonometric Graphs & Models · 13 topics

Amplitude of a Sinusoid [E]	Amplitude is |a|, half the vertical distance from crest to trough.
Period in Radians: 2π / b [M]	In radians the period of a sinusoid is 2π divided by b.
Period in Degrees: 360 / b [E]	In degrees the period of a sinusoid is 360 divided by b.
The Midline y = d [E]	The midline is y = d, the horizontal center the wave swings around.
Vertical Shift of a Sinusoid [E]	Adding d shifts the whole graph up (d > 0) or down (d < 0) by |d|.
Phase Shift of a Sinusoid [M]	Factor the b out: sin(bx − c) shifts right by c/b, not by c.
Period of y = tan(bx) [M]	Tangent repeats twice as fast as sine: its period is π / b.
Maximum & Minimum Values [M]	Max is midline + amplitude; min is midline − amplitude.
Amplitude from Max & Min [M]	Amplitude is half the gap between the highest and lowest values.
Midline from Max & Min [M]	The midline sits at the average of the highest and lowest values.
Modeling: Finding the Amplitude [M]	Turn a periodic phenomenon's high and low into an amplitude.
Modeling: Finding the Midline [M]	The model's midline is the center height or the average of high and low.
Modeling: Finding the Period [M]	The period is the time for one full cycle of the phenomenon.

Precalculus: Parametric & Polar Applications · 13 topics

Evaluating a Parametric Path [E]	Plug a value of t into each equation to locate the moving point.
Eliminating the Parameter: Lines [M]	Solve x = t + b for t, substitute, and read off slope and intercept.
Eliminating the Parameter: Parabolas [M]	A squared parameter eliminates into a quadratic in x - expand carefully.
Eliminating the Parameter: Circles [M]	x = a cos t, y = a sin t squares and adds to x² + y² = a².
Displacement Along a Parametric Path [M]	Displacement in a coordinate is its ending value minus its starting value.
Distance Between Two Positions [M]	The distance between two positions is √(Δx² + Δy²).
Polar to Rectangular: Exact Radicals [M]	At 30°, 45°, and 60° one coordinate carries an exact radical.
Polar to Rectangular: Whole Coordinates [M]	Half the special-angle conversions land on a plain rational coordinate.
Rectangular to Polar: the Radius [M]	The polar radius is the distance from the origin, √(x² + y²).
The Modulus of a Point [M]	A point's modulus is its polar radius: √(x² + y²).
The Circle r = a [E]	When r is a constant, every angle gives the same distance - a circle.
The Circle r = a cos θ [M]	r = a cos θ is an off-center circle of radius a/2 through the origin.
Classifying Polar Graphs [M]	Sort r = a, θ = c, r = a cos θ, and r = a cos(nθ) by their shapes.

Precalculus - Deep II · 37 topics

Trig Equations: a Solution in Radians [H]	Read the reference angle, place it in the right quadrants, and report the asked solution.
Trig Equations: Sum of the Solutions [H]	Add the two solutions on the interval - symmetry often makes the sum a tidy multiple of pi.
Quadratic Trig Equations: Counting Roots [H]	Solve the quadratic in sine or cosine, then count how many angles each root gives.
Half-Angle Exact Values [H]	cos(theta/2) = sqrt((1 + cos theta)/2); the sign follows the half-angle's quadrant.
Combining a Sine and Cosine: R sin(x + φ) [H]	a sin x + b cos x is one sinusoid with amplitude R = sqrt(a^2 + b^2).
The Tangent Sum & Difference Formula [H]	tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B).
The Law of Sines [H]	Sides are proportional to the sines of their opposite angles: a/sin A = b/sin B.
Law of Cosines: Finding a Side [H]	c^2 = a^2 + b^2 - 2ab cos C uses two sides and the included angle.
Law of Cosines: Finding an Angle [H]	Rearrange to cos C = (a^2 + b^2 - c^2)/(2ab) to recover an angle from three sides.
The Ambiguous (SSA) Case [H]	Compare the opposite side a to the height b sin A: it decides 0, 1, or 2 triangles.
Area from Two Sides and the Included Angle [H]	Area = (1/2) a b sin C - half the product of two sides times the sine of the angle between.
Polar Equations to Rectangular [H]	Multiply through by r and use r^2 = x^2 + y^2, r cos = x, r sin = y.
Rectangular Equations to Polar [H]	Substitute x = r cos, y = r sin; circles become r = a, and lines become r cos = k or theta = c.
Recognizing Polar Curves [H]	r = a(1 + cos) is a cardioid; r = a + b cos is a limacon; the nth-multiple angle marks a rose.
Work as a Dot Product [H]	Work is the dot product of force and displacement: W = Fx dx + Fy dy.
The Angle Between Two Vectors [H]	cos of the angle is the dot product over the product of the magnitudes.
Perpendicular & Parallel Vectors [H]	Perpendicular means the dot product is 0; parallel means one is a scalar multiple of the other.
Direction Angle of a Vector [H]	The direction angle points the vector; equal-magnitude components land on the 45-degree diagonals.
Parametric Ellipses [H]	x = a cos t, y = b sin t squares and adds to x^2/a^2 + y^2/b^2 = 1.
Speed Along a Parametric Line [H]	For linear motion the speed is sqrt((dx/dt)^2 + (dy/dt)^2), constant in time.
Projectile Height from Parametric Motion [H]	The height y = v0 t - 5 t^2 is a downward parabola; its vertex gives the peak.
Area of an Ellipse [H]	An ellipse encloses pi a b, the two semi-axes multiplied (a generalized pi r^2).
Matching a Conic to Its Equation [H]	Equal same-sign squares are a circle; unequal an ellipse; opposite signs a hyperbola; one square a parabola.
Polynomial End Behavior [H]	The leading term's degree parity and sign fix both tails of the graph.
Zeros & Their Multiplicities [H]	A factor's exponent is the zero's multiplicity; the total is the degree, and parity decides cross vs touch.
Slant (Oblique) Asymptotes [H]	When the numerator's degree is one more than the denominator's, long division gives a slant line.
Intercepts of a Rational Function [H]	x-intercepts are the numerator's zeros; the y-intercept is f(0).
Sum of an Arithmetic Series [H]	S_n = (n/2)(2 a_1 + (n - 1) d), or n times the average of the first and last terms.
Sum of a Finite Geometric Series [H]	S_n = a_1 (r^n - 1)/(r - 1) for a common ratio r not equal to 1.
Sum of an Infinite Geometric Series [H]	When |r| < 1 the series converges to a_1 / (1 - r).
The nth Term of a Sequence [H]	Arithmetic: a_1 + (n - 1) d. Geometric: a_1 r^(n - 1).
The Constant Term of a Binomial Expansion [H]	In (a x + c/x)^n the general term is C(n,k)(a x)^(n-k)(c/x)^k; the power of x is zero when k = n/2.
A Specific Term of a Binomial Expansion [H]	The coefficient of x^j in (x + a)^n is C(n, j) a^(n - j) - no need to expand fully.
Limits at a Removable Discontinuity [H]	Cancel the common factor, then substitute - the hole's height is the limit.
Limits at Infinity of Rational Functions [H]	Equal top and bottom degrees give the leading-coefficient ratio; a smaller top gives 0.
Estimating a Limit Numerically [H]	If both sides of a table close in on the same value, that value is the limit.
Limits from a Graph [H]	The limit watches the approach from both sides, not the plotted point's value.

Precalculus - Function Analysis & Transformations (Deep) · 43 topics

Operations on Functions at a Point [H]	Evaluate each function at the input first, then combine those two outputs with the indicated operation.
A Coefficient of the Product fg [H]	Collect every pair of terms whose exponents add to the requested power, then add those products.
Excluded Values of a Quotient f/g [H]	The quotient f/g is undefined wherever g is zero and wherever either function is already undefined.
The Domain of a Sum [H]	The domain of f + g is the overlap of the two domains: satisfy the radical condition and avoid the value that makes the denominator zero.
Evaluating a Composition [H]	Evaluate the inner function first, then apply the outer function to that output.
Solving a Composition Equation [H]	Undo the outer function first to learn what the inner output must be, then solve the inner equation.
Domain of a Rational Composition [H]	Exclude every input where the inner function is undefined and every input whose output is barred from the outer function's domain.
Domain of a Radical Composition [H]	Require the outer radicand to be non-negative, or strictly positive when the radical sits in a denominator, then solve that inequality for x.
Composition from a Table of Values [H]	Read the inner output from its column, then look that number up in the x column to read the outer output.
Decomposing h into f of g [H]	The inner function is the expression the outer operation is applied to; the outer function is that operation acting on a single placeholder.
Recovering the Outer Function [H]	Set u equal to the inner formula, solve for x in terms of u, and substitute to express h entirely in u; that expression is f.
A Value of the Inverse [H]	Finding f inverse of k means solving f(x) = k for x, since the inverse reverses the roles of input and output.
Verifying an Inverse Formula [H]	A candidate is the inverse exactly when composing it with f in either order returns x.
The Inverse of a Rational Function [H]	Set the rational expression equal to the target value, clear the denominator, and collect the x-terms on one side.
Inverses on a Restricted Domain [H]	Solving the quadratic equation gives two candidate inputs; the restricted domain selects which one is the inverse value.
Domain and Range of an Inverse [H]	An inverse swaps the two sets: the range of f inverse is the domain of f, and the domain of f inverse is the range of f.
Even, Odd, or Neither [H]	Replace x by negative x and compare: matching f gives an even function, matching the opposite of f gives an odd function, neither gives neither.
Using Symmetry to Find Values [H]	Symmetry transfers a known value across zero: an even function repeats it and an odd function reverses its sign.
Parity of Products and Compositions [H]	Track the sign that each factor or layer contributes when x is replaced by negative x, then multiply those signs together.
The Image of a Point [H]	Vertical constants act directly on the y-coordinate, while horizontal constants act on the x-coordinate in the opposite sense.
A Sequence of Transformations [H]	Translate each described step into the constant it contributes, keeping horizontal shifts inside the parentheses and vertical changes outside.
Building and Evaluating a Transformed Function [H]	Write the described sequence as an equation in the template A times the parent of B times x minus C, plus D, then substitute the input.
Extreme Values on a Closed Interval [H]	Compare the values at the endpoints with the value at the vertex, using the vertex only when it lies inside the interval.
How a Transformation Moves a Zero [H]	A zero survives every vertical stretch or reflection; only the horizontal replacement of x changes where it occurs.
Evaluating a Piecewise Function [H]	Decide which condition the input satisfies, then use only that branch's rule.
Making a Piecewise Rule Continuous [H]	Continuity at the boundary requires the two branch formulas to produce the same output there, which is one equation in the unknown constant.
The Corner of an Absolute Value Graph [H]	The corner occurs where the expression inside the absolute value equals zero, since that is where the branch definition switches.
Greatest-Integer Arithmetic [H]	The greatest integer function rounds toward negative infinity, so a negative non-integer moves down to the next integer.
A Step-Function Pricing Model [H]	Charges that jump at fixed intervals are modeled with a ceiling: any part of an interval counts as a whole one.
Average Rate of Change of a Rational Function [H]	Evaluate the function at both endpoints, subtract the outputs, and divide by the change in the input.
Comparing Average Rates from a Table [H]	Divide each change in output by its own change in input, then compare the resulting slopes rather than the raw changes.
Average Rate of Change in Context [H]	The secant slope carries the units of output per unit of input, so the computation answers a question about an average speed, rate, or yield.
Recovering a Missing Endpoint [H]	Write the average rate of change with the unknown endpoint in it, set that expression equal to the given rate, and solve.
Simplifying the Difference Quotient [H]	Expand f(x + h), subtract f(x) so every surviving term contains h, then divide each term by h.
The Difference Quotient of a Rational Function [H]	Combine the two rational outputs over a common denominator first; the numerator then contains a factor of h that cancels.
The End-Behavior Model of a Rational Function [H]	For large inputs only the leading terms matter, so the function behaves like the quotient of those two leading terms.
Where a Graph Crosses Its Horizontal Asymptote [H]	A horizontal asymptote describes end behavior only, so setting the function equal to that value and solving locates any crossing.
One-Sided Behavior at a Vertical Asymptote [H]	Test the sign of each factor at an input just to the chosen side of the asymptote; the combined sign decides which way the graph goes.
The Value Missing from the Range [H]	Solving y = f(x) for x fails at exactly one output, the height of the horizontal asymptote, so that value is missing from the range.
Area as a Function of One Variable [H]	Use the fencing or perimeter constraint to write the second dimension in terms of the first, then multiply the two dimensions.
The Open-Box Volume Model [H]	Cutting a square of side x from each corner shortens both base dimensions by twice x and makes x the height.
Composing a Geometric Model with Time [H]	Substitute the time-dependent radius into the geometric formula to obtain the quantity as a single function of time.
Revenue and Profit Models [H]	Revenue is price times quantity demanded, and profit subtracts both the per-item cost and the fixed cost from that revenue.

Precalculus - Trigonometric Functions, Identities & Equations (Deep) · 45 topics

Coterminal Angles in Radians [H]	Add or subtract whole multiples of 2 pi until the angle lands in the required window.
Degree and Radian Conversion Beyond the Special Angles [H]	Multiply degrees by pi/180 to get radians, and radians by 180/pi to get degrees.
Arc Length from a Degree Measure [H]	Arc length is s = r theta with theta in radians, so convert the degree measure first.
The Central Angle from an Arc [H]	Divide the arc length by the radius to get the central angle in radians.
Angular Speed in Radians per Second [H]	Angular speed is angle swept per unit time; one revolution is 2 pi radians.
Linear Speed on a Rotating Wheel [H]	Linear speed is v = r omega: the rim travels the radius times the angle swept.
Two Pulleys on One Belt [H]	A belt forces equal linear speeds, so r1 omega1 = r2 omega2 and the smaller pulley spins faster.
Coordinates of a Terminal Point [H]	The terminal point of theta on the unit circle is (cos theta, sin theta), signed by quadrant.
Recovering the Angle from a Terminal Point [H]	Read the reference angle from the size of the coordinates, then let their signs pick the quadrant.
Exact Secant, Cosecant, and Cotangent Values [H]	Take the reciprocal of the matching sine, cosine, or tangent value, then rationalize.
Locating the Quadrant from Two Signs [H]	Two sign conditions pin down one quadrant: all positive in I, sine in II, tangent in III, cosine in IV.
Combining Several Exact Values [H]	Evaluate each special value with its quadrant sign, then combine like radical terms.
Choosing b from a Required Period [H]	Period = 2 pi / b, so b = 2 pi divided by the required period.
Choosing c from a Required Phase Shift [H]	In sin(bx - c) the shift is c/b, so a shift of h needs c = b h.
Evaluating a Sinusoid Exactly [H]	Substitute, simplify the argument to a special angle, then scale by the amplitude and add the shift.
The End of One Full Cycle [H]	A cycle runs from argument 0 to argument 2 pi, so it ends at x = (c + 2 pi)/b.
Counting Crossings, Peaks, and Intercepts [H]	On 0 to 2 pi a sinusoid runs b full cycles, so features repeat b times.
A Ferris Wheel Model: Height at a Given Time [H]	Evaluate the height model by turning the elapsed time into a fraction of one revolution.
When a Model First Reaches an Extreme [H]	Maxima and minima alternate every half period, so step half a period from a known extreme.
Solving a Sinusoidal Model for a Time [H]	Write the depth as M + A cos(2 pi t / P), set it equal to the target, and solve for t.
Writing the Equation of a Sinusoidal Model [H]	Amplitude is half the range, the midline is the average of the extremes, and b is 2 pi over the period.
Asymptotes of a Tangent Graph [H]	Tangent is undefined, and its graph has a vertical asymptote, where its argument equals pi/2 plus a multiple of pi.
Counting Asymptotes of Secant and Cosecant [H]	Secant breaks where cosine is zero and cosecant breaks where sine is zero.
x-Intercepts of Tangent and Cotangent Graphs [H]	Tangent is zero where sine is zero; cotangent is zero where cosine is zero.
The Range Gap of a Cosecant or Secant Graph [H]	Since |csc| and |sec| are at least 1, the range is everything except the open interval of width 2|a| about d.
Principal Values of arcsin(sin x) and Friends [H]	An inverse function returns the angle in its own principal range, which need not be the angle you started with.
Compositions with an Inverse Trig Function [H]	Let the inverse name an angle, sketch its reference triangle, then read off the requested ratio.
Double Angles of an Inverse Trig Value [H]	Name the inverse value as an angle, find its sine and cosine, then apply the double-angle formula.
Domains and Ranges of the Inverse Trig Functions [H]	arcsin and arccos accept inputs in [-1, 1]; their outputs live in [-pi/2, pi/2] and [0, pi].
Complementary Sums of Inverse Values [H]	arcsin k + arccos k = pi/2 for every k in [-1, 1], and arctan k + arctan(1/k) = pi/2 for k > 0.
Secant from Tangent and Back [H]	Dividing sin^2 + cos^2 = 1 by cos^2 gives 1 + tan^2 = sec^2; by sin^2 it gives 1 + cot^2 = csc^2.
Simplifying a Trigonometric Fraction [H]	Replace a Pythagorean block with its single-function equivalent, then cancel.
Cofunction Identities [H]	Each function of an angle equals its cofunction of the complement: sin theta = cos(90 degrees - theta).
Identity or Conditional Equation? [H]	An identity holds for every defined value; a conditional equation holds only at particular angles.
Reading a Sum Formula Backwards [H]	Recognize the expanded pattern and compress it into one function of the combined angle.
Exact Tangent Values by Decomposition [H]	Split the angle into two special angles, apply tan(A + B), then rationalize the denominator.
The Tangent Double-Angle Formula [H]	tan 2 theta = 2 tan theta / (1 - tan^2 theta).
Half-Angle Values with Nested Radicals [H]	sin(theta/2) = plus or minus sqrt((1 - cos theta)/2), with the sign set by the half angle's quadrant.
Power-Reduction Formulas [H]	sin^2 theta = (1 - cos 2 theta)/2 and cos^2 theta = (1 + cos 2 theta)/2.
The Key Step in Verifying an Identity [H]	Verify by transforming one side alone: expand, write everything in sines and cosines, then combine.
Multiple-Angle Equations: a Requested Solution [H]	Solve for the whole argument bx over its own longer interval, then divide every solution by b.
Counting Solutions of a Multiple-Angle Equation [H]	Compressing the graph by a factor of b multiplies the number of solutions on a fixed interval by b.
Equations Needing a Double-Angle Substitution [H]	Rewrite the doubled angle in terms of the single angle, then factor the resulting equation.
The General Solution of a Trig Equation [H]	Add 2 k pi to each solution on one revolution, except for tangent, which repeats every pi.
Solving a Shifted Trig Equation [H]	Solve for the shifted argument first, then add the shift back and keep only the solutions in the interval.

Precalculus - Vectors, Polar Form, Parametrics & Matrices (Deep) · 44 topics

The Unit Vector in a Given Direction [H]	Dividing a vector by its own magnitude produces the unit vector pointing the same way.
Component Form from Magnitude and Direction [H]	A vector of magnitude m at direction angle theta has components m cos theta and m sin theta.
The Magnitude of a Scalar Multiple [H]	Scaling a vector by k multiplies its length by the absolute value of k.
The Magnitude of a Resultant Force [H]	Add the forces component by component, then take the length of the sum.
The Direction Angle of a Resultant [H]	Add the forces first, then read the direction angle of the sum from its components.
The Equilibrant of a Force System [H]	The equilibrant is the opposite of the resultant, so it has the same magnitude and reversed components.
Components from a Compass Bearing [H]	A bearing is measured clockwise from north, so the east component uses sine and the north component uses cosine.
Ground Speed and Wind in Navigation [H]	Ground velocity is air velocity plus wind velocity, and ground speed is the magnitude of that sum.
The Exact Angle Between Two Vectors [H]	Compute the cosine from the dot product and the magnitudes, then name the angle it belongs to.
The Magnitude of a Sum from the Dot Product [H]	Expanding |u + v|^2 as |u|^2 + 2(u dot v) + |v|^2 links lengths to the dot product.
The Scalar Projection onto a Vector [H]	The scalar projection of u onto v is the dot product divided by the length of v.
The Vector Projection and the Orthogonal Part [H]	The vector projection scales v by (u dot v) over the square of the length of v.
Weight Components on a Ramp [H]	On a ramp inclined theta, weight splits into w sin theta along the ramp and w cos theta into it.
Writing a Vector as a Combination of Two Others [H]	Matching components in a u + b v = w produces a two-by-two system for the coefficients.
Rectangular to Polar: the Angle [H]	The polar angle comes from the reference angle of |y/x| placed in the quadrant of the point.
Equivalent Polar Representations of a Point [H]	Adding 180 degrees to the angle reverses the sign of the radius, and adding 360 degrees changes nothing.
Where a Polar Curve Passes Through the Pole [H]	Set r equal to zero and solve the resulting trigonometric equation for theta.
Classifying a Limacon by the Ratio a/b [H]	For r = a plus or minus b times a cosine, the ratio a/b decides inner loop, cardioid, dimpled, or convex.
Counting the Petals of a Rose Curve [H]	A rose r = a cos(n theta) has n petals when n is odd and 2n petals when n is even.
Greatest and Least Distance from the Pole [H]	Since a sine or cosine ranges from -1 to 1, r = a plus or minus b trig ranges from a - b to a + b.
The Distance Between Two Polar Points [H]	Two radii and the angle between them form a triangle, so the law of cosines gives the distance.
The Argument of a Complex Number [H]	The argument is the direction angle of a + bi, found from the reference angle and the quadrant.
Trigonometric Form of a Complex Number [H]	Trigonometric form writes z as r(cos theta + i sin theta) with r the modulus and theta the argument.
Products in Trigonometric Form [H]	Multiplying complex numbers in trigonometric form multiplies the moduli and adds the arguments.
Quotients in Trigonometric Form [H]	Dividing complex numbers in trigonometric form divides the moduli and subtracts the arguments.
De Moivre's Theorem: Powers in Trigonometric Form [H]	Raising r(cos theta + i sin theta) to the nth power raises the modulus to the nth power and multiplies the argument by n.
De Moivre's Theorem in Rectangular Form [H]	Convert to trigonometric form, apply De Moivre, then convert the power back to a + bi.
The nth Roots of a Complex Number [H]	The n roots share the modulus r to the power 1/n and their arguments start at theta/n and step by 360/n degrees.
Eliminating the Parameter: Hyperbolas [H]	Secant and tangent parametrizations collapse to a hyperbola through sec squared minus tan squared equals one.
The Range of a Coordinate on a Parametric Path [H]	Because sine and cosine stay between -1 and 1, a coordinate a cos t + h stays between h - a and h + a.
Initial Velocity Components of a Projectile [H]	A launch speed v at angle theta splits into a horizontal v cos theta and a vertical v sin theta.
The Time of Flight of a Projectile [H]	Setting the height equation equal to zero and solving for t gives the moment the projectile lands.
The Horizontal Range of a Projectile [H]	The range is the horizontal velocity multiplied by the total time of flight.
The Height at a Given Horizontal Distance [H]	Solve the horizontal equation for the time at that distance, then substitute into the height equation.
Where a Parametric Path Crosses an Axis [H]	A path meets the y-axis where x(t) = 0 and the x-axis where y(t) = 0, so solve that equation for t first.
The Inverse of a 2x2 Matrix [H]	The inverse of [[a, b], [c, d]] swaps a and d, negates b and c, and divides everything by the determinant.
Singular Matrices: When No Inverse Exists [H]	A square matrix has an inverse exactly when its determinant is nonzero.
The 3x3 Determinant [H]	Expand along a row using cofactors: each entry multiplies the 2x2 determinant left after deleting its row and column.
Cramer's Rule for a 2x2 System [H]	Each variable equals the determinant of the coefficient matrix with that column replaced by the constants, divided by the determinant.
Cramer's Rule in a 3x3 System [H]	One variable of a three-variable system is a ratio of two 3x3 determinants.
Solving a Matrix Equation with an Inverse [H]	Multiplying both sides of A X = B by the inverse of A on the left isolates X.
The Dimensions of a Matrix Product [H]	A product AB exists only when the columns of A match the rows of B, and it has the rows of A and the columns of B.
Multiplying Matrices Beyond 2x2 [H]	The entry in row i and column j of a product is the dot product of row i of the left matrix with column j of the right.
Back-Substitution from Row-Echelon Form [H]	Row-echelon form solves the last variable immediately, and each earlier row then yields one more.

Prerequisite material - taught automatically when the diagnostic finds gaps

Arithmetic Foundations · 8 topics

Adding & Subtracting Whole Numbers	Multi-digit addition and subtraction.
Multiplication	Multiplying whole numbers.
Division	Dividing whole numbers.
Order of Operations	Parentheses first, then multiplication/division, then addition/subtraction.
Negative Numbers: Adding & Subtracting	Working with numbers below zero on the number line.
Negative Numbers: Multiplying & Dividing	Sign rules for products and quotients.
Exponents	Repeated multiplication in shorthand.
Square Roots	Undoing a square.

Fractions · 4 topics

Equivalent Fractions	Different fractions can name the same amount.
Simplifying Fractions	Reducing a fraction to lowest terms.
Multiplying Fractions	Multiply straight across.
Dividing Fractions	Multiply by the reciprocal.

Decimals, Percents & Ratios · 4 topics

Fractions ↔ Decimals	Converting between the two notations.
Percent of a Number	Percent means per hundred.
Percent Increase & Decrease	Applying a percent change to a quantity.
Ratios & Proportions	Two quantities that scale together.

Expressions & Equations · 7 topics

Evaluating Expressions	Substituting a value for a variable.
Combining Like Terms	Adding the coefficients of matching variable parts.
The Distributive Property	Multiplying across a sum.
One-Step Equations	Undoing a single operation.
Two-Step Equations	Undo addition/subtraction first, then multiplication.
Multi-Step Equations	Equations needing distribution or variables on both sides.
Linear Inequalities	Solving with <, >, ≤, ≥.

Linear Functions · 6 topics

The Coordinate Plane	Locating points with (x, y) pairs.
Slope of a Line	Rise over run between two points.
Slope-Intercept Form	y = mx + b describes a whole line.
Systems of Equations (Substitution)	Two equations, two unknowns.
Arithmetic Sequences	Add the same amount each step.
Geometric Sequences	Multiply by the same ratio each step.

Quadratics & Polynomials · 10 topics

Adding & Subtracting Polynomials	Combining polynomials by collecting like terms.
Multiplying Binomials (FOIL)	Expanding products of binomials.
Factoring Out the GCF	Undoing the distributive property.
Factoring Trinomials	Reversing FOIL: finding two numbers that multiply to c and add to b.
Special Factoring Patterns	Difference of squares and perfect-square trinomials.
Solving Quadratics by Factoring	Zero-product property: if a·b = 0 then a = 0 or b = 0.
Solving x² = k	Taking square roots of both sides - remembering ±.
Completing the Square	Turning any quadratic into a perfect square plus a constant.
The Quadratic Formula	x = (−b ± √(b² − 4ac)) / 2a solves any quadratic.
Vertex of a Parabola	The turning point at x = −b/2a.

Radicals & Exponentials · 5 topics

Product Rule for Exponents	Multiplying powers of the same base adds the exponents.
Quotient & Power Rules	Dividing powers subtracts exponents; a power of a power multiplies them.
Zero & Negative Exponents	Anything (nonzero) to the 0 power is 1; a negative exponent flips to a reciprocal.
Simplifying Radicals	Pulling perfect-square factors out of a square root.
Exponential Growth & Decay	Quantities that multiply by the same factor each time step: y = a·bᵗ.

Geometry · 8 topics

Angle Relationships	Vertical, complementary, and supplementary angle pairs.
Triangle Angle Sum	The three angles of a triangle always add to 180°.
The Pythagorean Theorem	In a right triangle, a² + b² = c².
Distance & Midpoint	Measuring segments in the coordinate plane.
Similar Triangles	Same shape, different size: corresponding sides are proportional.
Perimeter & Area	Measuring around and inside basic shapes.
Circles: Area & Circumference	C = 2πr and A = πr².
Special Right Triangles	The 45-45-90 and 30-60-90 side ratios.

Functions & Algebra II · 14 topics

Function Notation & Evaluation	Reading f(x) notation and plugging in inputs.
Domain & Range	Which inputs a function accepts, and which outputs it can produce.
Function Composition	Feeding one function's output into another: f(g(x)).
Inverse Functions	The function that undoes f: f⁻¹(b) is the input that f sends to b.
Transformations of Functions	How f(x − h) + k slides a graph around the plane.
Piecewise Functions	Functions defined by different rules on different intervals.
Systems by Elimination	Adding or subtracting equations to cancel a variable.
Polynomial Division	Dividing a polynomial by (x − a) with long or synthetic division.
Remainder & Factor Theorems	The remainder when p(x) is divided by (x − a) is simply p(a).
Zeros of Polynomials	Finding all the roots of a cubic by factoring it down.
End Behavior of Polynomials	Far from the origin, only the leading term matters.
Simplifying Rational Expressions	Factor top and bottom, then cancel the common factor.
Arithmetic Sequences	Sequences that grow by a constant difference each step.
Geometric Sequences	Sequences that grow by a constant ratio each step.

Limits & Continuity · 4 topics

Limits: Graphical & Numerical	What value a function approaches - which need not be the value it takes.
One-Sided Limits	Approaching from the left or right - and when the two disagree.
Infinite Limits & Vertical Asymptotes	A limit of +∞ or −∞ at a vertical asymptote; the sign of the shrinking denominator decides which.
Limits at Infinity	The value a function approaches as x goes to +∞ or −∞; for a rational function, compare the degrees of the top and bottom.

Infinite Series · 2 topics

Convergence of Sequences (BC)	A sequence converges if aₙ approaches a limit.
Geometric Series (BC)	Σarⁿ = a/(1−r) when |r| < 1.

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