Trigonometric and Polar Functions: every key term you need (+ practice quiz)
38 flashcard terms for AP Precalculus Unit 3, written to match the course framework. Study them here, then drill them as interactive flashcards, or test yourself with the 22-question quiz — free, no account needed.
A function whose values repeat at regular intervals: f(x + p) = f(x) for all x, where the smallest such positive p is the period.
Radian
The angle subtended at the center of a circle by an arc equal in length to the radius; a full circle is 2π radians and π radians = 180°.
Arc Length
s = rθ for a central angle θ measured in radians on a circle of radius r.
Standard Position
An angle with its vertex at the origin and initial ray along the positive x-axis; positive angles rotate counterclockwise.
Coterminal Angles
Angles in standard position that share the same terminal ray; they differ by integer multiples of 2π (or 360°).
Unit Circle
The circle x^2 + y^2 = 1 centered at the origin; a point at angle θ has coordinates (cos θ, sin θ).
Sine Function
sin θ is the y-coordinate of the unit-circle point at angle θ; for a circle of radius r it is y/r. Range is [−1, 1].
Cosine Function
cos θ is the x-coordinate of the unit-circle point at angle θ; for a circle of radius r it is x/r. Range is [−1, 1].
Tangent Function
tan θ = sin θ / cos θ = y/x, the slope of the terminal ray; undefined where cos θ = 0, giving vertical asymptotes at θ = π/2 + kπ.
Reference Angle
The acute angle between the terminal ray and the x-axis; trig values of any angle equal ± the values at its reference angle.
Special Angles
At π/6, π/4, π/3 the (cos, sin) values are (√3/2, 1/2), (√2/2, √2/2), (1/2, √3/2); memorize these and use symmetry for the rest of the circle.
Sinusoidal Function
Any function of the form f(θ) = a·sin(b(θ − c)) + d or the cosine equivalent; its graph is a wave with amplitude, period, phase shift, and midline.
Amplitude
|a| in a·sin(b(θ − c)) + d: half the distance between the maximum and minimum output values.
Midline
The horizontal line y = d halfway between a sinusoid's max and min; d is the vertical shift.
Period of a Sinusoid
2π/|b| for a·sin(b(θ − c)) + d; b compresses or stretches the graph horizontally by changing how fast the input cycles.
Phase Shift
The horizontal translation c in a·sin(b(θ − c)) + d; the graph of sine shifted right c units. Note cos θ = sin(θ + π/2).
Frequency
The number of cycles per unit of input, the reciprocal of the period: frequency = |b|/(2π).
Sinusoidal Concavity
A sinusoid is concave down where it is above its midline and concave up where it is below; inflection points occur at midline crossings, where the rate of change is greatest in magnitude.
sec θ = 1/cos θ, csc θ = 1/sin θ, cot θ = cos θ/sin θ; each has vertical asymptotes where its reciprocal function is zero.
Graph of Tangent
Period π, passes through the origin, increasing on each branch, with vertical asymptotes at odd multiples of π/2; it has no amplitude.
Pythagorean Identity
sin^2 θ + cos^2 θ = 1, from the unit circle equation; dividing gives 1 + tan^2 θ = sec^2 θ and 1 + cot^2 θ = csc^2 θ.
Sum and Difference Identities
sin(α ± β) = sin α cos β ± cos α sin β and cos(α ± β) = cos α cos β ∓ sin α sin β; used to find exact values and prove other identities.
Double-Angle Identities
sin 2θ = 2 sin θ cos θ; cos 2θ = cos^2 θ − sin^2 θ = 2cos^2 θ − 1 = 1 − 2sin^2 θ.
Even and Odd Trig Functions
cos(−θ) = cos θ (even), while sin(−θ) = −sin θ and tan(−θ) = −tan θ (odd).
Inverse Sine (arcsin)
sin^−1(x) returns the angle in [−π/2, π/2] whose sine is x; domain restriction makes sine one-to-one.
Inverse Cosine (arccos)
cos^−1(x) returns the angle in [0, π] whose cosine is x; domain of arccos is [−1, 1].
Inverse Tangent (arctan)
tan^−1(x) returns the angle in (−π/2, π/2) whose tangent is x; defined for all real x with horizontal asymptotes y = ±π/2.
Solving Trigonometric Equations
Isolate the trig expression, find the reference angle with an inverse function, use symmetry to find all solutions in one period, then add multiples of the period for the general solution.
Equivalent Trig Representations
Because of identities and periodicity, one function has many formulas: e.g., cos θ = sin(θ + π/2) = −cos(θ − π), so different-looking sinusoids may have identical graphs.
Polar Coordinates
A point (r, θ) located by directed distance r from the pole (origin) and angle θ from the polar axis; (r, θ), (r, θ + 2π), and (−r, θ + π) all name the same point.
Polar to Rectangular Conversion
x = r cos θ and y = r sin θ; going the other way, r^2 = x^2 + y^2 and tan θ = y/x (with quadrant checked).
Polar Circles
r = a is a circle of radius |a| centered at the pole; r = a cos θ and r = a sin θ are circles of diameter |a| passing through the pole.
Limaçon
r = a ± b cos θ or a ± b sin θ; a cardioid when a = b, an inner loop when a < b, dimpled when b < a < 2b, and convex when a ≥ 2b.
Rose Curve
r = a cos(nθ) or a sin(nθ); n petals if n is odd and 2n petals if n is even, each of length |a|.
Rate of Change in Polar Functions
When r is increasing on an interval of θ, the point moves away from the pole; when r is decreasing (even if negative), the point moves toward the pole. Distance from the pole is |r|.
Negative r Values
When r < 0, the point is plotted |r| units in the direction opposite θ; this is why r = 1 + 2cos θ has an inner loop.
Rate of rotation ω = Δθ/Δt in radians per unit time; linear speed of a point on a wheel of radius r is v = rω.
Modeling with Sinusoids
For periodic data (tides, temperature, Ferris wheels), find midline from (max + min)/2, amplitude from (max − min)/2, period from the repeat length, and phase shift from a known max, min, or midline crossing.