Every lesson in the Trigonometry slide course, in full text: 36 decks, 2347 slides.
10.1a Angles, Standard Position, and Degree MeasureAngles as amounts of rotation rather than as static corners: degree measure as a fraction of a revolution, the degree-minute-second system and both conversions, complementary and supplementary pairs, oriented angles whose sign records direction, standard position and the quadrant an angle terminates in, and coterminal angles as the family of rotations sharing one terminal side.
10.1b Radian Measure and the Degree-Radian BridgeRadian measure defined as arc length divided by radius, and why that ratio is the same for every circle: one radian is the angle cutting off an arc as long as the radius, a full revolution is two pi radians, and because a length over a length has no units, an angle in radians is a pure real number. Includes both conversion factors, standard position and coterminal angles restated in radians, and the wrapping of the number line onto the unit circle that identifies each real number with an angle.
10.1c Arc Length, Circular Motion, and Sector AreaThe formulas radian measure was built to produce: arc length as radius times angle with no conversion constant, sector area as half the radius squared times the angle, and the two velocities of circular motion — angular velocity as the rate the angle changes and linear velocity as the rate the position changes, linked by v equals r omega. Includes ordinary frequency, angular frequency and period, and the unit discipline that makes every one of these formulas either work or fail.
10.2a The Unit Circle: Cosine and SineCosine and sine defined as the two coordinates of the point where an angle's terminal side crosses the unit circle, rather than as ratios in a triangle. Covers reading the quadrantal values straight off the axes, deriving the values at pi over six, pi over four and pi over three from the two special right triangles together with the circle equation, the Pythagorean identity as the circle equation with its coordinates renamed, and recovering a missing coordinate from the identity plus quadrant information.
10.2b Reference Angles and the Quadrant SignsThe Reference Angle Theorem and what it buys: the acute angle a terminal side makes with the x-axis determines the size of both coordinates, and the quadrant supplies the two signs, so five memorised values generate all sixteen special points on the unit circle. Covers the four subtraction rules, the denominator shortcut in radian measure, angles built symmetrically from a given one, and the first trigonometric equations — where a single value produces two families of infinitely many solutions.
10.2c Beyond the Unit Circle: Circles of Any RadiusLifting the coordinate definition off the unit circle by similar triangles: on a circle of radius r the point at angle theta has coordinates r cosine theta and r sine theta, and conversely the cosine and sine can be read off any point on the terminal side as x over r and y over r. Includes the equations of circular motion as explicit functions of time, right triangle trigonometry recovered as the first-quadrant special case, and the domain and range of cosine and sine once they are read as functions of a real number.
10.3a The Six Circular Functions and the Fundamental IdentitiesThe four remaining circular functions defined from the same two coordinates: secant and cosecant as the reciprocals of cosine and sine, tangent and cotangent as their quotients, each with the restriction its denominator forces. Covers the geometric origin of the names tangent and secant, the reciprocal and quotient identities, the Generalized Reference Angle Theorem extending the reference-angle method to all six, the two further Pythagorean identities obtained by dividing the first through by a square, and the techniques for verifying an identity including Pythagorean conjugates.
10.3b Right Triangle Trigonometry and Angles of ElevationAll six circular functions lifted off the unit circle: expressed as ratios of x, y and r for any point on the terminal side, and as ratios of the three sides of a right triangle of any size. Covers reconstructing a point from a single function value plus a quadrant, the angle of inclination and the classic one- and two-sighting height problems, and closes with the domains and ranges of all six functions including the extended interval notation the book uses to write them.
10.4a Even-Odd, Sum and Difference, and Cofunction IdentitiesThe first half of the identity chapter, all of it descending from one distance-formula computation. Covers the even-odd identities and their proof by reflection across the x-axis, the difference identity for cosine proved from equal chords, the cofunction identities that fall out of it and explain the prefix co, the sum and difference identities for sine derived through the cofunction identities, and the tangent formula obtained by dividing. Includes finding exact values of non-special angles by decomposing them into special ones.
10.4b Double Angle, Power Reduction, and Half Angle FormulasThree identity families obtained without any new proof. Setting the two angles equal in the sum identities gives the double angle formulas, including the three equivalent forms of the cosine one; solving those backwards gives the power reduction formulas that trade a square for a doubled angle; and substituting theta over two into those and taking square roots gives the half angle formulas, whose plus-or-minus is settled by the quadrant of the half angle rather than of the original.
10.4c Product-to-Sum and Sum-to-Product FormulasThe last identity family of the chapter, in both directions. Adding and subtracting the sum and difference identities makes the mixed terms cancel, giving three product-to-sum formulas that turn a product of circular functions into a sum of first powers. Reversing them, with the two angles renamed as a half-sum and a half-difference, gives the sum-to-product formulas that turn a sum into a product — the form Section 10.7 needs, because only a product can be set to zero one factor at a time.
10.5a Graphs of Cosine and Sine: The SinusoidThe graphs of cosine and sine, and the four numbers that describe every transformation of them. Covers periodicity as the property that makes one cycle sufficient, the fundamental cycle and its five quarter marks, the quarter-mark method for graphing any sinusoid without tracking transformations one at a time, the identification of amplitude, period, phase shift and vertical shift from a formula and from a graph, and the recognition that a cosine plus a sine at the same frequency is itself a single sinusoid.
10.5b Graphs of Secant and CosecantThe first graphs in the course with vertical asymptotes, and every feature of them inherited from cosine and sine by reciprocation. Covers where the asymptotes come from and how to determine which way each branch runs, the two values reciprocation leaves fixed, the range being everything of absolute value at least one, the absence of any amplitude, and the practical method of drawing the parent sinusoid first and reciprocating it point by point.
10.5c Graphs of Tangent and CotangentThe last two graphs, and the two that behave least like the rest. Both have period pi rather than two pi, both have range all real numbers, and each branch climbs or falls monotonically between consecutive asymptotes instead of turning around. Covers the proof that the period really is pi using the tangent sum identity, the geometric reason behind it, the different fundamental cycles and quarter marks the two functions use, and graphing transformed versions.
10.6a Arcsine, Arccosine, and ArctangentNone of the six circular functions is one-to-one, so none has an inverse until its domain is restricted. Covers why the restriction is necessary and how the intervals are chosen, the resulting properties of arccosine, arcsine, arctangent and arccotangent, the asymmetric cancellation rules that make arccos of cos x fail to be x outside a specific interval, and the substitution technique that turns any composition of a circular function with an inverse one into an ordinary triangle problem.
10.6b Inverses of Secant and CosecantThe two circular functions whose inverses are genuinely not standardised. Covers why the secant is awkward to invert at all — its range has a gap, so the inverse's domain does too — and then presents both competing restrictions in full: the trigonometry-friendly one that keeps the arccosine's interval, and the calculus-friendly one chosen to make the tangent of an arcsecant a single formula rather than a piecewise one. The same worked examples are answered under each, so the differences are visible rather than described.
10.6c Calculators and Solving Equations with Inverse FunctionsThe inverse functions put to work. Covers approximating the three inverses a calculator does not provide, finding the domain and range of a transformed inverse function by tracking key points, applications where a measured ratio must be converted into an angle, and the main point of the section: solving equations whose values are not special angles, using an inverse function to name the reference angle exactly as a square root names an irrational solution.
10.7a Solving Trigonometric EquationsThe chapter's techniques applied together. Every trigonometric equation is reduced to one of four basic forms, and the reduction is the whole skill: substituting for a compound argument and unwinding correctly, factoring a quadratic in disguise, using identities to match the functions or the arguments, converting a sum into a product so it can be set to zero factorwise, and collapsing a cosine plus a sine into a single sinusoid. Includes the rule that a trigonometric expression is never divided out, and how to count the solutions falling in a stated window.
10.7b Trigonometric Inequalities and Extended DomainsThe last lesson of Chapter 10. An inequality asks where one side exceeds the other, and the answer is a union of intervals rather than a list of points. Covers the sign diagram technique with its two kinds of landmark, choosing test values in awkward gaps, using the same machinery to find the domain of any function built from circular functions, the strategy of solving on one period and translating, and equations and inequalities in the inverse trigonometric functions, which behave in the opposite way to everything else in the chapter.
11.10 Parametric EquationsThe last idea of the course: let both coordinates depend on a third variable, so a curve becomes the path of a moving point. Covers sketching from a parametrization by reading each coordinate's behaviour separately, eliminating the parameter by substitution or by a trigonometric identity, the standard recipes for parametrizing graphs, segments and ellipses, and the two adjustments that reverse an orientation or delay a start. Closes with the cycloid, a curve that has no usable equation in x and y at all.
11.1a Applications of SinusoidsChapter 11 opens by putting Section 10.5 to work. The sine form of the sinusoid is adopted once and for all, and each of its four parameters is given a physical meaning that word problems state in ordinary language. Covers deriving a sinusoid from a physical setup such as a rotating wheel, the distinction between period, ordinary frequency and angular frequency, fitting a sinusoid to measured data by taking the baseline and amplitude from the extremes, and locating the phase from the position of a maximum.
11.1b Harmonic MotionA mass on a spring oscillates sinusoidally without going round anything. Covers Hooke's law and the equilibrium position from which all displacement is measured, the equation for free undamped harmonic motion with omega determined by the apparatus and the amplitude and phase by the initial conditions, and then the three ways the idealisation is relaxed: damped motion with a decaying envelope, resonance where forcing at the natural frequency makes the envelope grow, and beats where forcing at a different frequency produces a slow modulation that sum-to-product exposes.
11.2 The Law of Sines and the Ambiguous CaseThe first tool for a triangle with no right angle. Covers the Law of Sines and its proof by dropping an altitude, the two configurations it handles unambiguously, and the Angle-Side-Side case where the same data may describe no triangle, exactly one, or two genuinely different ones — decided by comparing the given opposite side against the altitude c sine alpha. Closes with the area formula in terms of two sides and their included angle.
11.3 The Law of Cosines and Heron's FormulaThe law that finishes the job the Law of Sines cannot start: Side-Angle-Side and Side-Side-Side. Proves it by dropping a triangle into standard position and applying the distance formula, shows how it generalises the Pythagorean Theorem, and explains why the sign of the cosine makes it immune to the ambiguous case. Closes with Heron's Formula for the area of a triangle from its three sides alone.
11.4a Polar Coordinates: Plotting and Multiple NamesA second address system for the plane, built from a pole and a polar axis rather than two number lines. Covers plotting from a directed distance and a rotation, what a negative first coordinate and a negative angle each mean, and the property that characterises exactly when two polar pairs name the same point — the reason a point has infinitely many polar names and only one rectangular one.
11.4b Converting Between Polar and Rectangular FormTheorem 11.7 and what it does and does not settle. Converts points in both directions, showing why the arctangent alone cannot determine the angle and why every conversion into polar form starts with a sketch. Then converts equations both ways, contrasting the mechanical rectangular-to-polar substitution with the awkward reverse, and examining what multiplying by r and squaring both sides do to a solution set.
11.5a Graphs of Polar EquationsHow to draw a polar curve without converting it to rectangular form. Establishes the Fundamental Graphing Principle for polar equations and the two shapes given by a constant coordinate, then develops the book's two-plane method: sketch r against theta on ordinary axes and read that sketch as instructions for the polar plane. Applies it to limacons, a four-petalled rose, and a lemniscate, with the negative-r intervals handled explicitly throughout.
11.5b Polar Intersections and Polar RegionsWhere two polar curves meet, and why solving the two equations simultaneously does not answer that question. Establishes the four-step guidelines — sketch and check the pole, solve directly, shift by a full turn, flip the sign and half-turn — through the circle-and-rose example whose eight intersection points are found four by solving and four only by the last step. Closes with a pair of equations that turn out to describe the same curve, and with describing regions of the plane by inequalities on r and theta.
11.6a Rotation of AxesThe conics again, this time tilted. Derives the rotation equations from polar coordinates and the sum formulas, uses them to convert both points and equations into a rotated frame, and finds the angle that eliminates the cross term — the one whose cotangent of twice it equals A minus C over B. Closes with the discriminant, which classifies a general second-degree equation as hyperbola, parabola or ellipse without performing any rotation at all.
11.6b The Polar Form of ConicsOne definition covering all four conics: the set of points whose distance to a fixed focus is a constant multiple of the distance to a fixed directrix. In polar coordinates with the focus at the pole, that definition becomes a single equation with one parameter, the eccentricity, whose value against 1 decides between ellipse, parabola and hyperbola. Covers the four standard forms and their directrices, reading a given equation by normalising it first, and the rotated general form that collapses all four into one.
11.7a Modulus, Argument, and the Polar Form of a Complex NumberComplex numbers as points in a plane, and the polar description of those points. Defines the modulus as the distance from the origin and the argument as the set of angles that reach the number, singling out the principal argument in the interval from negative pi to pi. Establishes the properties of both, then assembles the polar form: the modulus times cosine theta plus i sine theta, abbreviated cis, and converts in both directions.
11.7b DeMoivre's Theorem and the nth Roots of a Complex NumberWhat the polar form is for. Establishes that multiplication multiplies moduli and adds arguments, that division divides and subtracts, and that powers follow by induction as DeMoivre's Theorem. Reads the product rule geometrically as a stretch followed by a rotation, then solves the root equation to show that every non-zero complex number has exactly n distinct nth roots, evenly spaced around a circle at the vertices of a regular polygon.
11.8a Vectors: Components, Addition, and Scalar MultiplicationQuantities carrying both a magnitude and a direction, and the algebra they obey. Defines a vector as a directed segment, establishes that position is irrelevant, and records it as a pair of components by subtracting the tail from the head. Adds vectors tip to tail and componentwise, showing the two agree, applies the resultant to a navigation problem via the Law of Cosines, and closes with the properties of vector addition and of scalar multiplication.
11.8b Magnitude, Direction, and Unit VectorsHow long a vector is and which way it points, extracted from its components. Defines the magnitude as the distance from tail to head and the direction as the unit vector pointing the same way, establishing that every vector is its magnitude times its direction. Uses that to resolve a vector into components from a stated size and bearing, introduces normalising and the principal unit vectors, and closes with a static equilibrium problem solved by setting a vector sum to zero.
11.9 The Dot Product and ProjectionA product of two vectors returning a number. Defines it componentwise, establishes its algebraic properties, then proves via the Law of Cosines that it equals the product of the magnitudes times the cosine of the angle between them. From that follow the angle formula, the test for perpendicularity, the projection of one vector onto another with its unique parallel-plus-perpendicular decomposition, and the definition of work as a dot product of force and displacement.
Session 1: Prerequisite Algebra and the Unit CircleA running start for a college trigonometry course: the four algebra skills the course leans on hardest — complex fractions, rationalising, the disguised-quadratic pattern and graph transformations — then angle measure in degrees and radians, arc length and sector area, both special triangles rebuilt from scratch, and the unit circle with reference angles and quadrant signs.
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