Precalculus 3: Trigonometry

Navigate section two, a crash course in Euclidean geometry, to refresh or prepare for pi and trig in right triangles, with optional skip or revisit later.

Explore the foundations of euclidean geometry—primitive notions, axioms, definitions, and theorems—covering points, lines, plane, betweenness, congruence, and postulates relevant to trigonometry.

Define angle as the figure formed by two rays sharing a vertex, and denote angles with three letters, exploring standard position, coterminal and quadrantal angles.

Explore how angles relate by measure and by position, defining congruent, vertical, supplementary, complementary, adjacent, and linear-pair angles. Note that alternate angles are discussed in video 15.

Define triangles and their standard notation, then explain congruence tests with sss, sas, and asa, illustrating why some rules fail and how to construct triangles.

Explore the fifth postulate and how two parallel lines cut by a third line create interior, exterior, alternate, and corresponding angles; prove angle relations and parallelogram properties using congruent triangles.

Explain why the triangle's interior angles sum to 180 degrees and classify triangles as acute, right, or obtuse. Show exterior angles equal the sum of remote interior angles.

The video defines area for generalized polygons using axioms, derives the rectangle area as x times y for rational sides, and builds understanding via unit squares and induction.

Derive the area of a triangle from base and altitude using area axioms, and apply the 1/2 base times height formula via rectangle division; explore right triangles and orthocenter.

Explore Thales’ theorem and the side-splitting proportions of similar triangles to derive parallelism and the basic proportionality theorem, laying the groundwork for defining sine, cosine, and tangent in right triangles.

Master right triangles with the Pythagorean theorem, hypotenuse and legs, and congruence rules for right triangles, plus contrapositive and converse forms and the 3-4-5 example.

Explore the properties of equilateral and isosceles triangles, prove that an isosceles triangle's height bisects the base, and derive the height of an equilateral triangle using the Pythagorean theorem.

Explore inscribed and central angles in a circle and learn that an inscribed angle is half of the central angle subtending the same arc, with the diameter subtending 90 degrees.

Discover how every triangle has a unique inscribed circle and circumscribed circle, find their centers via angle and segment bisectors, and learn compass-based constructions.

Explore regular polygons as convex figures with equal sides and angles, derive interior angle measure as 180 minus 360 over n, and show the sum of interior angles equals 180(n-2).

Explore an early egyptian approximation of pi from the Rhind Mathematical Papyrus by estimating the disk area, yielding about 3.16 r^2, and note pi’s irrational and transcendental nature.

Archimedes explores pi via perimeters of inscribed regular polygons, deriving a recursive method to approximate the circle’s circumference as polygon sides double, from hexagon to 96 sides.

Explore the intuitive link between pi and the area of a disk. Visualize inscribed polygons that approach a rectangle with sides r and pi r, illustrating pi r^2.

Explore how Eudoxus’ method of exhaustion proves the disk area equals half the circumference times the radius, linking pi to circle area through inscribed and circumscribed polygons.

Explore how geometry underpins trigonometric functions, introducing sine, cosine, and tangent through proportions, ratios, and similar triangles, and revisit the Pythagorean theorem for right triangles.

Define sine, cosine, tangent and their reciprocals from a right triangle, explain six possible ratios, and establish cofunction identities that relate angles alpha and its complement.

Practice applying the six trigonometric functions to right triangles, using opposite, adjacent, and hypotenuse definitions, the Pythagorean theorem, and reciprocal relationships.

Practice defining the six trigonometric functions for angle alpha in right triangles using the Pythagorean theorem, opposite and adjacent legs, and cofunction identities.

Construct right triangles to realize sine, cosine, and tangent values for a given x, compute the other functions, and apply these constructions to trig substitutions in calculus.

Compute sine, cosine, and tangent of 45 degrees using a unit square and the 45-45-90 triangle; derive sin = cos = 1/sqrt(2) and tan = 1, and discuss rationalizing denominators.

Explain how plotting sine yields cosine via cofunction identities, and show cosine's decreasing, positive 0 to 1 behavior for acute angles along with symmetry to sine and related identities.

Compute exact values of sine, cosine, and tangent for key angles using geometry to build the sine curve, and apply cofunction identities.

Derive sine, cosine, and tangent values for 30 and 60 degrees using an equilateral triangle of side one. Verify results with Pythagorean identity and plot points on the sine curve.

Demonstrate the sum identities for sine and cosine using geometry method two with two unit right triangles, proving sin(a+b) and cos(a+b) and deriving tan(a+b).

Derive sine and cosine double-angle formulas and tangent from identities, then establish power-reduction formulas cos^2 alpha = (1+cos 2alpha)/2 and sin^2 alpha = (1−cos 2alpha)/2 via Pythagorean identity.

Use half-angle formulas to compute sine and cosine of 15 degrees, then deduce 75 degrees as a complementary angle, illustrating sign considerations for acute angles.

Apply the sum identities to compute sine and cosine of 75 degrees from 45 and 30 degrees. Conclude with sine and cosine of 15 degrees via complementary angles.

Explore a geometric method to compute sine and cosine of 18 and 72 degrees using isosceles and similar triangles, height bisectors, and complementary angles that lead to a quadratic equation.

Compute sine and cosine of 54 and 36 degrees geometrically using isosceles triangles, height, and double angle and Pythagorean identities, and connect to 18 degrees and the golden ratio.

Derives sine and cosine of triple angles using sum and double-angle formulas, showing the triple-angle identities. Presents sin(3α)=3sinα−4sin^3α and cos(3α)=4cos^3α−3cosα.

Using double and triple angle formulas, the lecture derives sin 18° by equating sin 3x with cos 2x, yields a polynomial, and finds sin 18° = (√5−1)/4.

Complete the table of exact values by deriving sine and cosine of half an angle, 27 degrees, from 54-degree values using double-angle formulas and the Pythagorean identity.

Apply the same trick to 15 degrees, derive cosine and sine of 15 degrees using double angle formulas and the Pythagorean identity, solving a linear system for x and y.

Explore different radical forms of sine and cosine for 15 degrees and verify their equality and the Pythagorean identity, using the square of sum and radical arithmetic.

Explore tangent half-angle formulas expressing sine, cosine, and tangent of alpha in terms of tan(alpha/2) through algebraic and geometric derivations on the unit circle.

Compute sine, cosine, and tangent values for 30, 45, and 60 from equilateral triangles or squares. Illustrate how trigonometric formulas and the sine graph apply to non-acute angles and domains.

Explore inverse trigonometric functions—arcsin, arccos, and arctan—through a geometric view, covering domains, ranges, and 1-to-1 conditions with sine, cosine, and tangent. Learn why radians matter for plotting these inverses.

Explore solving triangles using trigonometry, focusing on right triangles: determine all angles and sides from one angle, two sides, or leg-hypotenuse data, aided by Pythagorean theorem and inverse trig.

Learn to solve right triangles using the Pythagorean theorem and trigonometric functions, with or without approximations, and determine when problems yield one, two, or no solutions.

Master elementary problem solving in precalculus trigonometry by practicing with right triangles, using tangent, sine, and cosine to find angle alpha, verify pythagorean identities, and compare exact versus approximate results.

This section argues for measuring angles in radians, not degrees, to define circular functions as real-valued, enabling sine, cosine, arctangent with real inputs and Taylor approximations.

Define standard position angles in the Cartesian plane and identify coterminal and quadrantal angles. Relate radians to unit circle coordinates and use reference angles to derive sine, cosine, and tangent.

Extend sine and cosine from acute angles to all real numbers via the unit circle and a rotating point, linking to x^2+y^2=1 and the pythagorean identity.

Extend geometric definitions of sine and cosine from acute angles to all real angles using a circular framework. Map their graphs and identities on the unit circle for problem solving.

Wrap the number axis around the unit circle to represent any real angle using radians, coterminal angles, reference angles, and multiples of pi along the t axis.

Define sine and cosine as the x and y coordinates on the unit circle for arc length t in radians, highlighting co-terminal angles and degree-radian conversions.

Explore dynamic definitions of sine and cosine via the unit circle, where a point moves along arc length t, mapping x=cos t and y=sin t to their graphs.

Define all six circular functions from sine and cosine of a real variable using the unit circle, establish domain restrictions, and link to reciprocal, quotient, and pythagorean identities.

Explore how reference angles share sine and cosine values across quadrants using unit circle symmetries. Apply the reference angle theorem and complementary formulas to reuse exact values.

Explore periodic functions by plotting the fractional part of x, reveal how its graph repeats with period one, and explain why circular functions are not injections.

Explore the sine function, its domain, range, zeros, and periodicity, and learn how its graph on the unit circle reveals its properties.

Learn how the cosine function derives from the sine graph by shifting left by pi half, revealing its domain, range, zeros, even symmetry, and monotonic intervals.

Define the cotangent as cosine divided by sine and view it as the tangent's reciprocal when nonzero; its domain excludes multiples of pi with period pi and range real numbers.

Explore the secant function: its definition as the reciprocal of cosine, domain exclusions at pi/2 plus k pi, and the secant graph with asymptotes, range, and period 2pi.

Explore the etymology of the six circular functions and how unit circle geometry links sine, cosine, tangent, cotangent, secant, cosecant, co-functions, and the Pythagorean identity.

Demonstrates solving circular functions exercise 5 by using pythagorean identities and quadrant signs to derive sine, cosine, tangent and reciprocals from one given value, including the 7-24-25 triple.

Derive the cosine sum identity for general angles using a unit circle and distance formula, presenting two proofs in orthogonal coordinate systems for the cosine of alpha plus beta.

Derive the sine sum identity from the cosine sum identity using complementary angle relations, showing sine alpha plus beta equals sine alpha cosine beta plus cosine alpha sine beta.

Derive the cosine difference identity using the cosine sum formula and the even/odd properties of cosine and sine, obtaining cos alpha cos beta + sin alpha sin beta.

Graph transformations introduce vertical and horizontal shifts, reflections, and stretches of trig graphs, with examples showing how changes to f and to the input affect domain and range.

Learn to scale any nonzero pair of real numbers into cosine and sine of the same angle by dividing by their shared length, enabling standard form and plotting.

Rewrite a sin x + b cos x as a scaled sine with phase shift; amplitude √(a^2+b^2) and shift −β, with cos β = a/√(a^2+b^2) and sin β = b/√(a^2+b^2).

Rewrite a linear combination of sine and cosine as a scaled sine with a phase shift, then verify amplitude and vertical shift through exercises.

Reframe the function as 13 sin(x+alpha) + 12 using a 5-12-13 triangle, then show the max is 25 and min is -1 when sin(x+alpha) equals 1 or -1.

Explore even odd identities for trigonometric functions, showing cosine and secant are even and sine, tangent, cosecant, cotangent are odd, with domain restrictions and tangent and cosine exercises.

Explore sum and difference identities for sine, cosine, and tangent by solving exercise five with alpha in quadrant one and beta in quadrant two, using pythagorean, reciprocal, and quotient identities.

Apply double angle formulas to power reduction and derive half-angle formulas, then compute exact tangent and cosine values across quadrants using the unit circle and reference angles.

Derives three product-to-sum formulas to convert cos a cos b into a sum, sin a sin b into a difference, and sin a cos b into a sum of sines.

Explore how product-to-sum formulas transform products of cosines and sines into sums, using even and odd identities to solve exercises and prepare for advanced trigonometry applications.

Apply sum-to-product formulas to exercise 9, using cofunction identities to handle mixed sine and cosine cases and derive concise multi-angle expressions like sine of one degree.

Explore the monotonicity of sine in the first quadrant, proving sine is increasing on zero to pi over two via sum-to-product, and linking to inverse trig and arc sine.

Demonstrate the monotonicity of the cosine in the first quadrant by proving it decreases on [0, pi/2], extend via symmetry to intervals, and link to sine, tangent, and inverse functions.

Explore arc sine concepts in preparation for arc sine exercise ten, derive cosine from sine via Pythagorean identity, and compute sine and cosine of double angles for acute quadrant one.