### 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\mathrm{cos}\phantom{\rule{0.3em}{0ex}}\theta $ and $r=a\pm b\mathrm{sin}\phantom{\rule{0.3em}{0ex}}\theta ,$ where $\frac{a}{b}=1$

- convex limaҫon
- a type of one-loop limaçon represented by $r=a\pm b\mathrm{cos}\phantom{\rule{0.3em}{0ex}}\theta $ and $r=a\pm b\mathrm{sin}\phantom{\rule{0.3em}{0ex}}\theta $ such that $\frac{a}{b}\ge 2$

- De Moivre’s Theorem
- formula used to find the $n\text{th}$ power or
*n*th 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\mathrm{cos}\phantom{\rule{0.3em}{0ex}}\theta $ and $r=a\pm b\mathrm{sin}\phantom{\rule{0.3em}{0ex}}\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\mathrm{cos}\phantom{\rule{0.3em}{0ex}}\theta $ and $r=a\pm b\phantom{\rule{0.3em}{0ex}}\mathrm{sin}\phantom{\rule{0.3em}{0ex}}\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}\mathrm{cos}\phantom{\rule{0.3em}{0ex}}2\theta $ and ${r}^{2}={a}^{2}\mathrm{sin}\phantom{\rule{0.3em}{0ex}}2\theta ,$ $a\ne 0$

- 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\mathrm{cos}\phantom{\rule{0.3em}{0ex}}\theta $ and $r=a\pm b\mathrm{sin}\phantom{\rule{0.3em}{0ex}}\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 $x=r\mathrm{cos}\phantom{\rule{0.3em}{0ex}}\theta ,\phantom{\rule{1em}{0ex}}y=r\mathrm{sin}\phantom{\rule{0.3em}{0ex}}\theta ,$ 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\mathrm{cos}\phantom{\rule{0.3em}{0ex}}n\theta $ and $r=a\mathrm{sin}\phantom{\rule{0.3em}{0ex}}n\theta ;$ 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