Key Terms

Key Terms

altitude

a perpendicular line from one vertex of a triangle to the opposite side, or in the case of an obtuse triangle, to the line containing the opposite side, forming two right triangles

ambiguous case

a scenario in which more than one triangle is a valid solution for a given oblique SSA triangle

Archimedes’ spiral

a polar curve given by $r=\theta .$ When multiplied by a constant, the equation appears as $r=a\theta .$ As $r=\theta ,$ the curve continues to widen in a spiral path over the domain.

argument

the angle associated with a complex number; the angle between the line from the origin to the point and the positive real axis

cardioid

a member of the limaçon family of curves, named for its resemblance to a heart; its equation is given as $r=a\pm b\cos \theta$ and $r=a\pm b\sin \theta ,$ where $\frac{a}{b}=1$

convex limaҫon

a type of one-loop limaçon represented by $r=a\pm b\cos \theta$ and $r=a\pm b\sin \theta$ such that $\frac{a}{b}\ge 2$

De Moivre’s Theorem

formula used to find the $n\text{th}$ power or nth roots of a complex number; states that, for a positive integer $n,z^n$ is found by raising the modulus to the $n\text{th}$ power and multiplying the angles by $n$

dimpled limaҫon

a type of one-loop limaçon represented by $r=a\pm b\cos \theta$ and $r=a\pm b\sin \theta$ such that $1<\frac{a}{b}<2$

dot product

given two vectors, the sum of the product of the horizontal components and the product of the vertical components

Generalized Pythagorean Theorem

an extension of the Law of Cosines; relates the sides of an oblique triangle and is used for SAS and SSS triangles

initial point

the origin of a vector

inner-loop limaçon

a polar curve similar to the cardioid, but with an inner loop; passes through the pole twice; represented by $r=a\pm b\cos \theta$ and $r=a\pm b\sin \theta$ where $a<b$

Law of Cosines

states that the square of any side of a triangle is equal to the sum of the squares of the other two sides minus twice the product of the other two sides and the cosine of the included angle

Law of Sines

states that the ratio of the measurement of one angle of a triangle to the length of its opposite side is equal to the remaining two ratios of angle measure to opposite side; any pair of proportions may be used to solve for a missing angle or side

lemniscate

a polar curve resembling a figure 8 and given by the equation $r^2=a^2\cos 2\theta$ and $r^2=a^2\sin 2\theta ,$ $a\ne 0$

magnitude

the length of a vector; may represent a quantity such as speed, and is calculated using the Pythagorean Theorem

modulus

the absolute value of a complex number, or the distance from the origin to the point $\left(x,y\right);$ also called the amplitude

oblique triangle

any triangle that is not a right triangle

one-loop limaҫon

a polar curve represented by $r=a\pm b\cos \theta$ and $r=a\pm b\sin \theta$ such that $a>0,b>0,$ and $\frac{a}{b}>1;$ may be dimpled or convex; does not pass through the pole

parameter

a variable, often representing time, upon which $x$ and $y$ are both dependent

polar axis

on the polar grid, the equivalent of the positive x-axis on the rectangular grid

polar coordinates

on the polar grid, the coordinates of a point labeled $\left(r,\theta \right),$ where $\theta$ indicates the angle of rotation from the polar axis and $r$ represents the radius, or the distance of the point from the pole in the direction of $\theta$

polar equation

an equation describing a curve on the polar grid.

polar form of a complex number

a complex number expressed in terms of an angle $\theta$ and its distance from the origin $r;$ can be found by using conversion formulas $x=r\cos \theta ,y=r\sin \theta ,$ and $r=\sqrt{x^2+y^2}$

pole

the origin of the polar grid

resultant

a vector that results from addition or subtraction of two vectors, or from scalar multiplication

rose curve

a polar equation resembling a flower, given by the equations $r=a\cos n\theta$ and $r=a\sin n\theta ;$ when $n$ is even there are $2n$ petals, and the curve is highly symmetrical; when $n$ is odd there are $n$ petals.

scalar

a quantity associated with magnitude but not direction; a constant

scalar multiplication

the product of a constant and each component of a vector

standard position

the placement of a vector with the initial point at $\left(0,0\right)$ and the terminal point $(a,b),$ represented by the change in the x-coordinates and the change in the y-coordinates of the original vector

terminal point

the end point of a vector, usually represented by an arrow indicating its direction

unit vector

a vector that begins at the origin and has magnitude of 1; the horizontal unit vector runs along the x-axis and is defined as $v_1=\langle 1,0\rangle$ the vertical unit vector runs along the y-axis and is defined as $v_2=\langle 0,1\rangle .$

vector

a quantity associated with both magnitude and direction, represented as a directed line segment with a starting point (initial point) and an end point (terminal point)

vector addition

the sum of two vectors, found by adding corresponding components