Trigonometry Mastered

Understand the hierarchy of number sets—from natural numbers (N) and integers (Z) through rational and irrational numbers to real, imaginary, and complex numbers.

Explore the most graphing calculator tool, learn to graph functions, compare online and offline modes, and use derivatives, integrals, and parlor graphing features, plus apps and teaching resources.

Explore the coordinate plane formed by two perpendicular axes, the x axis and the y axis. Each point has an x and a y coordinate locating it in quadrants.

Explore trigonometry through three sections: the unit circle approach, the right triangle approach, and proofs and identities, starting with the unit circle.

the unit circle at the origin with radius one is described by x^2 + y^2 = 1, and you can shift circles using (x-h)^2+(y-k)^2=r^2 while testing points.

Master terminal points on the unit circle by traveling from (1,0) ccw or cw to angles like pi/6, pi/3, and pi/4, obtaining coordinates such as (0,1) and (±1/√2, ±1/√2).

determine the reference number, the shortest distance from a unit circle point to the x axis, and use it to locate terminal points in each quadrant.

Learn how trigonometric functions are defined from the unit circle by the terminal point, precisely. Sine equals y-coordinate, cosine equals x-coordinate, tangent equals y over x; cosecant, secant, cotangent.

learn how to evaluate sine, cosine, and tangent on the unit circle across quadrants using the add sugar to coffee memory trick, reference angles, and even/odd properties.

Learn to graph sine and cosine using the unit circle, with attention to x and y components and their cycles. Derive amplitude and period from coefficients and recognize periodic behavior.

Explore sine and cosine graph transformations, detailing amplitude and period changes, vertical shifts, and horizontal (phase) shifts, with examples of shifting and graphing over a full cycle.

Explore the tangent and cotangent graphs, their pi-periods, vertical asymptotes, and transformations. Analyze how shifts, scaling, and odd function properties shape these graphs.

Learn to graph secant and cosecant by applying horizontal shifts to the reciprocals of cosine and sine, identify vertical asymptotes and periods, and sketch one complete period.

Explore the domain and range of sine, cosine, and tangent, and see how vertical shifts and scaling alter their graphs.

Explain how to determine domain and range for sine, cosine, and tangent, including ranges and asymptotes illustrated by examples such as 3 sin(5x) minus 2 and sin(x-2) plus 8.

Master the domain and range of sec, csc, and cot by analyzing vertical asymptotes, horizontal scaling, and amplitude changes, with cotangent covering all real numbers and secant/cosecant excluding (-1,1).

Learn to determine domain and range for cotangent and cosecant functions, including transformed arguments, by locating vertical asymptotes and applying range rules.

Explore trigonometry through the right angle triangle, outlining angles and the functions used to find or evaluate distances.

Explore angles formed by rays with a common vertex, including the initial side and terminal side, and learn to measure and convert between degrees and radians.

Master standard position: the initial side on the positive x axis with the vertex at the origin. Identify coterminal angles by adding or subtracting 360 degrees, and recognize four quadrants.

Explore the six trigonometric ratios in a right triangle, defining sine, cosine, and tangent and their reciprocals cosecant, secant, and cotangent using opposite, adjacent, and hypotenuse.

Master right triangle trig by applying SOH CAH TOA to find sine, cosine, tangent and their reciprocals using opposite, adjacent, and hypotenuse; practice with example problems.

Learn to derive sine, cosine, and tangent for 30, 45, and 60 degrees using the Pythagorean theorem, memorize, and extend to 0, 90, 120, 180, 270, and 360 by graphing.

Apply sine, cosine, and other trig ratios to solve right triangles quickly, using hypotenuse and leg relationships and the Pythagorean theorem with 30 and 45 degree examples.

Identify reference angles as the smallest angle between the terminal side and the x axis, and use them to determine sine and cosine values for various angles.

Learn to evaluate trig functions from a point on the terminal side by forming a right triangle with x, y, and r = sqrt(x^2+y^2), use quadrant signs and reference angles.

Explore how inverse trigonometric functions arise from inverse functions by restricting sine, cosine, and tangent to specific domains. Learn the corresponding arcsine, arccosine, and arctangent with their domains and ranges.

Master the law of sines to solve triangles when given two sides and an angle, using sin ratios to find missing angles and recognize the ambiguous case.

Learn to apply the law of cosines to find sides or angles in a triangle using two sides and the included angle, with step-by-step examples and practice.

Compute triangle area from three sides via Heron's formula (semi perimeter) and from two sides with the included angle via sine. Includes 3-4-5 and 5-4-120° examples.

Master analytic trigonometry by exploring equations, identities, and proofs involving trigonometric functions. Develop fast formula recall through repeated writing and use the attached formula document to practice.

Explore fundamental trigonometric identities, including reciprocal, pythagorean, even and odd, and cofunction identities, to solve equations, simplify expressions, and prove relationships.

Learn to simplify and prove trig expressions using identities, convert to sine and cosine, and apply conjugates and cofunction identities through worked examples.

Master the addition and subtraction formulas for sine, cosine, and tangent, with examples that show how to compute angles like 15 and 75 degrees using sums and differences.

Learn to rewrite a sin x + b cos x as k sin(x+phi) with k = sqrt(a^2+b^2) and cos phi = a/k, sin phi = b/k; see examples.

Master double angle formulas for sin 2x, cos 2x, and tan 2x, including sin 2x = 2 sin x cos x and cos 2x = 2 cos^2 x - 1.

Learn and apply half-angle formulas for sine, cosine, and tangent, including plus/minus signs by quadrant, with worked examples like sin(pi/8) and cos(105 degrees) to solve problems.

Master formulas for lowering powers of sine, cosine, and tangent, rewriting higher powers in terms of first powers.

Learn sum-to-product formulas that convert sums and differences of sines and cosines into products, such as sin x + sin y and cos x + cos y.

Explore product-to-sum formulas that convert products of sines and cosines into sums, with examples like cos a cos b and sin a sin b, and practice via a companion document.

Explore polar coordinates, using distance from the origin and an angle to locate points, convert between polar and rectangular coordinates, and handle negative r and theta.

Explore converting between polar and rectangular coordinates using x = r cos theta, y = r sin theta, and r = sqrt(x^2 + y^2), with tan theta = y/x.

Explore converting polar and rectangular equations using x = r cos theta and y = r sin theta, derive r^2 = x^2 + y^2, and tan theta = y/x.

Learn to graph polar equations on a dedicated polar grid using r and theta, with examples like r=2 and f(x)=2+2cos x, and cos(2x) rose patterns.

Master symmetry in polar graphs by identifying symmetry about the polar axis, the pole, and the line theta = pi/2, then reflect half the graph.

Explore polar curves: circles from r = a, circles shifted by sine or cosine, spirals, limacons (cardioid and inner loop), and rose curves r = a sin(nθ) or a cos(nθ).

Graph complex numbers on the complex plane using the real and imaginary axes, in the context of polar coordinates, and locate points like 3+4i and i while practicing basic operations.

Compute the modulus of a complex number z = a + bi as the distance from the origin by sqrt(a^2 + b^2); this yields a real, positive value.

Convert complex numbers to polar form using modulus and the argument, then express z as r(cos theta + i sin theta) by identifying r and theta and the proper quadrant.

Master multiplying and dividing complex numbers in polar form by multiplying the modules and adding the arguments, or dividing the modules and subtracting the arguments, using cosign and sign components.

Learn to raise a complex number to a positive integer power using de moivre's theorem. Convert to polar form, multiply the argument by the power, and raise the modulus.

Learn to find nth roots of a complex number by converting to polar form and applying r^(1/n) [cos((θ+2πk)/n) + i sin((θ+2πk)/n)] for k = 0 to n-1, with examples.

Celebrate finishing the course and solidify concepts by revisiting lectures, then ask questions and share feedback; explore related math courses like graph theory, precalculus, calculus, and linear algebra.