Everything here starts with a circle. Follow a point around the unit circle to build sine, cosine, and tangent, use them to model anything that repeats, and then switch to the polar coordinate system, where angles and radii replace left-right and up-down.
Episode Guide — Season 3
Tides, clocks, and traffic all repeat. A relationship is periodic when outputs show a repeating pattern over equal-length input intervals, and the period is the length of one full cycle.
Put an angle in standard position on the unit circle and measure it in radians. The sine, cosine, and tangent come straight from the point where the angle's terminal ray meets the circle.
On a circle of radius r, the point at angle θ is (r cos θ, r sin θ). Use isosceles right and equilateral triangles to find exact values at multiples of π/4 and π/6.
Unwrap the unit circle and the sine and cosine graphs appear. Both oscillate between −1 and 1, tracking the vertical and horizontal displacement of a point as it travels around the circle.
Sine and cosine are the parent sinusoidal functions. Describe them with period, frequency, amplitude, and midline, and recognize that sine is odd while cosine is even.
In a·sin(b(θ + c)) + d, each constant does one job. Read amplitude, period, phase shift, and vertical shift straight off the equation, and build the function from them.
Model a periodic situation. Maxima and minima give you the amplitude and vertical shift, the spacing between peaks gives the period, and a known point gives the phase shift. Or use sinusoidal regression.
Tangent is the slope of the terminal ray. It has period π, vertical asymptotes wherever cosine is zero, and its transformations work a little differently than sine and cosine.
Arcsine, arccosine, and arctangent turn a trig value back into an angle. Because trig functions are periodic, each is only invertible after you restrict its domain.
Use inverse trig functions to solve equations and inequalities, then account for periodicity to find every solution. In a context, the domain often limits the answers.
The reciprocal functions secant, cosecant, and cotangent each have vertical asymptotes where their parent function is zero. Learn their graphs, ranges, and domains.
Rewrite trig expressions using the Pythagorean identity and the sine and cosine sum identities, and use those forms to solve equations and verify other identities.
Locate a point with a radius and an angle instead of left-right and up-down. Convert between polar and rectangular coordinates, and see complex numbers as points in a plane.
In r = f(θ), the input is an angle and the output is a signed radius. Graph polar functions on a restricted domain and compare them with their rectangular graphs.
Is a point moving toward the origin or away from it? Use the sign of r and whether it's increasing or decreasing to find out, and use average rate of change to describe how the radius changes per radian.