Trigonometric and Polar Functions

Periodic function ● (core concept) — A function whose output values repeat in a pattern over successive equal-length input intervals.

Period ● (core concept) — The smallest positive value k such that f(x + k) = f(x) for all x in the domain; the function's behavior is then determined by any interval of width k. It can be estimated from data by finding where the output pattern begins to repeat.

Standard position ● (core concept) — An angle is in standard position when its vertex is at the origin and one ray lies along the positive x-axis; the other ray is the terminal ray. Positive measures rotate counterclockwise, negative clockwise; angles sharing a terminal ray differ by an integer number of revolutions.

Radian measure ● (core concept) — The measure of an angle in standard position as the ratio of the arc length it subtends to the circle's radius. On a unit circle (radius 1), one radian is subtended by an arc of length 1.

Sine ● (core concept) — For an angle in standard position, the sine is the ratio of the vertical displacement of the terminal-ray point P from the x-axis to the distance OP; on the unit circle it is the y-coordinate of P.

Cosine ● (core concept) — For an angle in standard position, the cosine is the ratio of the horizontal displacement of P from the y-axis to the distance OP; on the unit circle it is the x-coordinate of P.

Tangent ● (core concept) — For an angle in standard position, the tangent is the slope of the terminal ray, when it exists: the ratio of P's y-coordinate to its x-coordinate on the unit circle, equivalently sin θ/cos θ.

Coordinates of a point on a circle ● (core concept) — For an angle of measure θ in standard position and a circle of radius r centered at the origin, the terminal-ray point is P = (r cos θ, r sin θ).

Exact trigonometric values ● (core concept) — The cosine and sine of angles that are multiples of π/4 and π/6 (with terminal rays off the axes) can be found exactly using the geometry of isosceles right and 30-60-90 (equilateral) triangles, attending to the signs for the terminal ray's quadrant.

Sine function ● (core concept) — f(θ) = sin θ gives, on the unit circle, the y-coordinate (vertical displacement from the x-axis) of the terminal-ray point; its domain is all real numbers and its outputs oscillate between −1 and 1.

Cosine function ● (core concept) — f(θ) = cos θ gives, on the unit circle, the x-coordinate (horizontal displacement from the y-axis) of the terminal-ray point; its domain is all real numbers and its outputs oscillate between −1 and 1.

Sinusoidal function ● (core concept) — Any function formed by additive and multiplicative transformations of sin θ; both sine and cosine are sinusoidal, with cos θ = sin(θ + π/2). A sinusoid's graph alternates between concave down and concave up as inputs increase. Sine is odd (symmetric about the origin) and cosine is even (symmetric over the vertical axis).

Period and frequency ● (core concept) — The period and frequency of a sinusoidal function are reciprocals. For sin θ and cos θ the period is 2π and the frequency is 1/(2π).

Amplitude ● (core concept) — Half the difference between a sinusoidal function's maximum and minimum values (1 for sin θ and cos θ).

Midline ● (core concept) — The horizontal line midway between a sinusoid's maximum and minimum values: their average (arithmetic mean); y = 0 for sin θ and cos θ.

Phase shift ● (core concept) — A horizontal translation of a sinusoidal graph. For g(θ) = sin(θ + c), the phase shift is −c units — the graph shifts left when c is positive.

Parameters of a sinusoidal function ● (core concept) — In y = a·sin(b(θ + c)) + d (a ≠ 0, b ≠ 0): amplitude |a|, period 2π/|b|, a vertical shift d of the midline from y = 0, and a phase shift of −c. The same holds for cosine, since cosine is a phase shift of sine by −π/2 units.

Sinusoidal regression ● (core concept) — A sinusoidal model fitted to periodic data with technology, built from estimated key values — period and frequency from consecutive maxima or minima, amplitude and vertical shift from the maximum and minimum, phase shift by comparing actual and model input-output pairs — or from a sinusoidal regression.

Tangent function ● (core concept) — f(θ) = tan θ gives the slope of the terminal ray, equal to sin θ/cos θ where cos θ ≠ 0. It has period π (slopes repeat every half revolution), vertical asymptotes at θ = π/2 + kπ (integers k) where cos θ = 0, and it increases — changing from concave down to concave up — between consecutive asymptotes.

Transformed tangent function ● (core concept) — y = a·tan(b(θ + c)) + d is a vertical dilation by a (a reflection over the x-axis if a < 0), has period π/|b| (a reflection over the y-axis if b < 0), shifts the line containing its points of inflection vertically by d, and phase-shifts by −c.

Inverse trigonometric functions ● (core concept) — Arcsine, arccosine, and arctangent (sin⁻¹, cos⁻¹, tan⁻¹) swap the inputs and outputs of sine, cosine, and tangent: the input is a value in the trig function's range and the output is interpreted as an angle measure. Because trig functions are periodic, they must have restricted domains to be invertible.

Restricted domains for inverse trig ● (core concept) — Sine is restricted to [−π/2, π/2], cosine to [0, π], and tangent to (−π/2, π/2) to define arcsine, arccosine, and arctangent respectively.

Secant ● (core concept) — sec θ = 1/cos θ, where cos θ ≠ 0. Its graph has vertical asymptotes where cos θ = 0 and range (−∞, −1] ∪ [1, ∞).

Cosecant ● (core concept) — csc θ = 1/sin θ, where sin θ ≠ 0. Its graph has vertical asymptotes where sin θ = 0 and range (−∞, −1] ∪ [1, ∞).

Cotangent ● (core concept) — cot θ = 1/tan θ (tan θ ≠ 0), equivalently cos θ/sin θ (sin θ ≠ 0). Its graph has vertical asymptotes where tan θ = 0 and decreases between consecutive asymptotes.

Pythagorean identity ● (core concept) — sin²θ + cos²θ = 1, from applying the Pythagorean theorem to the point (cos θ, sin θ) on the unit circle. It can be rewritten in other forms, such as tan²θ = sec²θ − 1.

Sine sum identity ● (core concept) — sin(α + β) = sin α cos β + cos α sin β.

Cosine sum identity ● (core concept) — cos(α + β) = cos α cos β − sin α sin β.

Difference and double-angle identities ● (core concept) — The sine and cosine sum identities also serve as difference identities (β = −α) and double-angle identities (β = α). Known identities and algebraic properties can be used to verify further identities.

Polar coordinate system ● (core concept) — A coordinate system built on circles centered at the origin and lines through the origin; the positive x-axis is the polar axis. A point is located by (r, θ): θ is the measure of an angle in standard position whose terminal ray passes through the point, and r is the signed radial displacement from the origin along — or opposite, for negative r — that ray. The same point can be represented many ways with combinations of positive and negative r and θ values.

Polar–rectangular conversion ● (core concept) — From (r, θ) to (x, y): x = r cos θ, y = r sin θ. From (x, y) to (r, θ): r = √(x² + y²) and θ = arctan|y/x| for x > 0, or θ = arctan|y/x| + π for x < 0, as given in the CED.

Complex number in polar form ● (core concept) — A complex number is a point in the complex plane: rectangular coordinates (a, b) give a + bi; polar coordinates (r, θ) give (r cos θ) + i(r sin θ).

Polar function ● (core concept) — r = f(θ): input values are angle measures and output values are radii; the graph is the set of input-output pairs (f(θ), θ). The distance from the origin is |r|.

Average rate of change in polar functions ● (core concept) — Over an interval of θ, the ratio of the change in signed radius values to the change in θ: the rate at which the signed radius changes per radian. It can be used to estimate function values within the interval.

Relative extrema in polar graphs ● (core concept) — When a polar function changes from increasing to decreasing (or vice versa) on an interval, it has a relative extremum corresponding to a point relatively closest to or farthest from the origin: if r is positive and increasing, or negative and decreasing, the distance from the origin increases; if r is positive and decreasing, or negative and increasing, the distance decreases.