Math III – Glossary | Open Up HS Math (CCSS), Family

A–F

acute angle

An angle whose measure is between $0 °$ and $90 °$ .

$∠ A B C$ is an acute angle.

acute triangle

A triangle with three acute angles.

Angles $A$ , $B$ , and $C$ are all acute angles.

Triangle $A B C$ is an acute triangle.

Ambiguous Case of the Law of Sines

The Ambiguous Case of the Law of Sines occurs when we are given SSA information about the triangle. Because SSA does not guarantee triangle congruence, there are two possible triangles.

To avoid missing a possible solution for an oblique triangle under these conditions, use the Law of Cosines first to solve for the missing side. Using the quadratic formula to solve for the missing side will make both solutions become apparent.

amplitude

The height from the midline (center line) to the maximum (peak) of a periodic graph. Half the distance from the minimum to the maximum values of the range.

For functions of the form $y = A \sin b θ$ or $y = A \cos b θ$ , the amplitude is $| A |$ .

angle of rotation in standard position

To represent an angle of rotation in standard position, place its vertex at the origin, the initial ray oriented along the positive $x$ -axis, and its terminal ray rotated $θ$ degrees counterclockwise around the origin when $θ$ is positive and clockwise when $θ$ is negative. Let the ordered pair $(x, y)$ represent the point where the terminal ray intersects the circle.

angular speed

Angular speed is the rate at which an object changes its angle in a given time period. It can be measured in $(\frac{degrees}{second})$ . Typically measured in $(\frac{radians}{second})$ .

arc length

The distance along the arc of a circle. Part of the circumference.

Equation for finding arc length:

$s = r θ$

Where $r$ is the radius and $θ$ is the central angle in radians.

argument of a logarithm

See logarithmic function.

asymptote

A line that a graph approaches, but does not reach. A graph will never touch a vertical asymptote, but it might cross a horizontal or an oblique (also called slant) asymptote.

Horizontal and oblique asymptotes indicate the general behavior of the ends of a graph in both positive and negative directions. If a rational function has a horizontal asymptote, it will not have an oblique asymptote.

Oblique asymptotes only occur when the numerator of $f (x)$ has a degree that is one higher than the degree of the denominator.

base of a logarithm

See logarithmic function (logarithm).

bimodal distribution

A bimodal distribution has two main peaks.

The data has two modes.

See also: modes.

binomial

A polynomial with two terms.

binomial expansion

When a binomial with an exponent is multiplied out into expanded form.

Example:

${(a + b)}^{2} = (a + b) (a + b) = 1 a^{2} + 2 a b + 1 b^{2}$

Pascal’s triangle (shown) can be used to find the coefficients in a binomial expansion. Each row gives the coefficients to ${(a + b)}^{n}$ , starting with $n = 0$ . To find the binomial coefficients for ${(a + b)}^{n}$ , use the $n th$ row and always start with the beginning variable raised to the power of $n$ . The exponents in each term will always add up to $n$ . The binomial coefficients for ${(a + b)}^{5}$ are $1$ , $5$ , $10$ , $10$ , $5$ , and $1$ — in that order or $1 a^{5} + 5 a^{4} b + 10 a^{3} b^{2} + 10 a^{2} b^{3} + 5 a b^{4} + 1 b^{5}$

Central Limit Theorem (CLT)

This theorem gives you the ability to measure how much your sample mean will vary, without having to take any other sample means to compare it with.

The basic idea of the CLT is that with a large enough sample, the distribution of the sample statistic, either mean or proportion, will become approximately normal, and the center of the distribution will be the true parameter.

clockwise / counterclockwise

clockwise: Moving in the same direction, as the hands on a clock move.

counterclockwise: Moving in the opposite direction, as the hands on a clock move.

closure

A set is closed (under an operation) if and only if the operation on any two elements of the set produces another element of the same set.

cluster sample

See sample.

common logarithm

A logarithm with base $10$ , written $\log x$ , which is shorthand for $\log_{10} x$ .

composition of functions

The process of using the output of one function as the input of another function.

$f (x) = 3 x - 1; g (x) = 5 x + 1$

Replace $x$ with $5 x + 1$ .

$f (g (x)) = 3 (5 x + 1) - 1$

$f (g (x)) = 15 x + 3 - 1 = 15 x + 2$

conjugate pair

A pair of numbers whose product is a nonzero rational number.

The numbers $(a + \sqrt{b})$ and $(a - \sqrt{b})$ form a conjugate pair.

The product of $(a + \sqrt{b}) (a - \sqrt{b}) = a^{2} - {(\sqrt{b})}^{2} = a^{2} - b$ , a rational number.

continuous compound interest

Continuously compounded interest means that the account constantly earns interest on the amount of money in the account at any time, which includes the principal and the interest earned previously.

control group

The control group is used in an experiment as a way to ensure that your experiment actually works. It is a baseline group that receives no treatment or a neutral treatment. To assess treatment effects, the experimenter compares results in the treatment group to results in the control group.

convenience sample

See sample.

coterminal angles

Two angles in standard position that share the same terminal ray but have different angles of rotation.

The diagram shows a positive rotation ( $blue$ ) of ray $A C$ from $B$ through $F$ to $C$ . The dotted arc ( $green$ ) shows a negative rotation of ray $A C$ from $B$ through $D$ to $C$ .

The two angles are coterminal.

cross-section of a solid

The face formed when a three-dimensional object is sliced by a plane.

cubic function

a polynomial of degree $3$ . The parent function is $y = x^{3}$ .

decomposition of functions

Undoing a composite function in terms of its component parts.

degree of a polynomial

The power of the term that has the greatest exponent.

density

In science, density describes how much space an object or substance takes up (its volume) in relation to the amount of matter in that object or substance (its mass). If an object is heavy and compact, it has a high density. If an object is light and takes up a lot of space, it has a low density.

$density = \frac{mass}{volume}$

Density can also refer to how many people are crowded into a small area or how many trees are growing in a small space or a large space. In that sense it is a comparison of compactness to space.

$density = \frac{how many}{space or area}$

disc or disk

See solid of revolution.

distribution curve

A graph of the frequencies of different values of a variable in a statistical distribution.

dividend

See division.

division

With polynomials:

division algorithm for polynomials

If $f (x)$ and $d (x) \neq 0$ are polynomials such that the degree of $d (x) \leq$ the degree of $f (x)$ , there exists unique polynomials $q (x)$ and $r (x)$ such that

where the degree of $r (x) <$ the degree of $d (x)$ .

If $r (x) = 0$ , then $d (x)$ divides evenly into $f (x)$ , making $d (x)$ a factor of $f (x)$ .

divisor

See division.

domain

The set of all possible $x$ -values which will make the function work and will output real $y$ -values. A continuous domain means that all real values of $x$ included in an interval can be used in the function.

Choosing a smaller domain for a function is called restricting the domain. The domain may be restricted to make the function invertible.

Sometimes the context will restrict a domain.

Other terms that refer to the domain are input values and independent variable.

double angle identities

See trigonometric identities.

edge / face / vertex of a 3-D solid

Edge: The line that is the intersection of two planes.

Face: A flat surface on a $3$ -D solid.

Vertex: (pl. vertices) Each point where two or more edges meet; a corner.

elapsed time

The time that has passed since the position of the rider was at the farthest right position on the wheel (standard position with initial ray along the positive $x$ -axis).

end behavior

The behavior of a function $y = f (x)$ for $x$ -values that are very large (approaching $\infty$ ) and very small (approaching $- \infty$ ).

even function

See function: even.

experiment

In an experiment, researchers separate the participants into a control group and a treatment group, and manipulate the variables to try to determine cause and effect. One of the key components of an experiment is that individuals are assigned to treatments and the results are compared.

explicit equation

Relates an input to an output.

Example: $f (t) = 4 t + 1$ ; $t$ is the input and $f (t)$ is the output.

The explicit equation is also called a function rule, an explicit formula, or explicit rule.

extraneous solution

A derived solution to an equation that is invalid in the original equation.

factor

Factor (verb): To factor a number means to break it up into numbers that can be multiplied together to get the original number.

Example: Factor $75$ : $75 = 3 \times 25$ , or $15 \times 5$ , or $3 \times 5 \times 5$

Factor (noun): a whole number that divides exactly into another number. In the example above $3$ , $5$ , $15$ , and $25$ are all factors of $75.$

In algebra factoring can get more complicated. Instead of factoring a number like $75$ , you may be asked to factor an expression like $15 a^{2} b$ .

The numbers $3$ and $5$ and the variables $a$ and $b$ are all factors. The variable $a$ is a factor that occurs twice.

factor of a polynomial

$(x - r)$ is a factor of the polynomial function $y = P (x)$ if dividing $P (x)$ by $x - r$ leaves no remainder.

frequency

The number of times the event occurred in an experiment or study.

frequency distribution curve, frequency polygon

A frequency distribution curve “smooths out the bumps” in a frequency distribution with a theoretical curve that shows how often an experiment will produce a particular result.

frustum

The part of a solid such as a pyramid or a cone that remains after cutting off a top portion with a plane parallel to the base.

function: even, odd

A function $f (x)$ is classified as an odd function if $f (- θ) = - f (θ)$ . Example: $f (x) = x^{3}$ is an example of an odd function. The graph of an odd function is symmetric with respect to the origin. This means it can be rotated $180 °$ and still look the same.

A function $f (x)$ is classified as an even function if $f (- θ) = f (θ)$ . Example: $f (x) = x^{2}$ is an even function. The graph of an even function reflects across the $y$ -axis.

G–L

horizontal asymptote

A line that the graph approaches but does not reach. Exponential functions have a horizontal asymptote. The horizontal asymptote is the value the function approaches as $x$ either gets infinitely larger or smaller. An asymptote is an imaginary line, but it is often shown as a dotted line on the graph.

$f (x) = 2^{x}$

As $x$ gets smaller, the graph of $f (x)$ approaches the horizontal asymptote, $y = 0$ .

$h (x) = 2^{- x}$

As $x$ gets larger, the graph of $h (x)$ approaches the horizontal asymptote, $y = 0$ .

$g (x) = 2^{x} - 3$

As $x$ gets smaller, the graph of $g (x)$ approaches the horizontal asymptote, $y = - 3$ .

M–R

margin of error

The margin of error is a statistic expressing the amount of random sampling error in the results of a survey. The larger the margin of error, the less confidence one should have that a poll result will reflect the result of a survey of the entire population.

maximum / minimum

Maximum is the point at which a function’s value is greatest.

Minimum is the point at which a function’s value is the least.

midline of a trigonometric function

A horizontal axis that is used as the reference line about which the graph of a periodic function oscillates. The equation of the midline is $y = k$ , where $k$ is the vertical translation of the function.

mode(s)

The measure of central tendency for a one-variable data set that is the value(s) that occurs most often.

Types of modes include: uniform (evenly spread- no obvious mode), unimodal (one main peak), bimodal (two main peaks), or multimodal (multiple locations where the data is relatively higher than others).

See measures of central tendency.

modulus

The modulus of the complex number $a + b i$ is $| a + b i | = \sqrt{a^{2} + b^{2} .}$ This is the distance between the origin $(0, 0)$ and the point $(a, b)$ in the complex plane.

For two points in the complex plane, the distance between the points is the modulus of the difference of the two complex numbers. The formula looks a lot like the formula for finding the distance between two points.

Example: Given two complex numbers:

$a + b i$ and $c + d i$ , the distance between them is

$d = \sqrt{{(c - a)}^{2} + {(d - b)}^{2}}$

Find the distance between $1 + 3 i$ and $- 2 - 1 i$ .

$d = \sqrt{{(- 2 - 1)}^{2} + {(- 1 - 3)}^{2}} = \sqrt{9 + 16} = 5$

multiplicity

The multiplicity of each zero is the number of times that its corresponding factor appears. If $p (x) = (x + 2) (x - 1) (x - 1) (x - 1)$ , the zeros or roots of $p (x)$ are $x = 2$ multiplicity $1$ and $x = 1$ multiplicity $3$ .

The multiplicity of a root affects the shape of the graph of a polynomial. If a root of a polynomial has odd multiplicity, the graph will cross the $x$ -axis at the root.

Graph of $p (x) = (x + 2) (x - 1) (x - 1) (x - 1)$

$x = 2$ multiplicity $1$ and $x = 1$ multiplicity $3$

If a root of a polynomial has even multiplicity, the graph will touch the $x$ -axis at the root but will not cross the $x$ -axis.

Graph of $q (x) = (x + 2) (x - 1) (x - 1)$

$x = 2$ multiplicity $1$ and $x = 1$ multiplicity $2$

natural logarithm

A logarithm with base $e$ , written $\ln x$ , which is shorthand for $\log_{e} x$ .

normal distribution

An arrangement of a data set in which most values cluster in the middle of the range and the rest taper off symmetrically toward either extreme. Features of a normal distribution include:

A normal distribution is symmetric.
The mean, median, and mode are equal in a normal distribution.
The frequency curve of a normal distribution is symmetric.
A normal curve has points of inflection at $\pm 1$ standard deviation from the mean.
A normal distribution has a single mode.

$68 %$ of the distribution will be $\pm 1$ standard deviations from the mean. $95 %$ of the distribution will be within $2$ standard deviations from the mean. $99.7 %$ of the distribution will be within $3$ standard deviations from the mean.

observational study

a study in which the researcher simply observes the subjects without interfering.

In this type of study, researchers observe the behavior of the participants/subjects without trying to influence it in any way so they can learn about the parameter of interest.

obtuse angle / obtuse triangle

odd function

See function: odd.

one-to-one function

A function $f$ is said to be one-to-one if no two elements of the domain of $f$ correspond to the same element in the range of $f$ . If no horizontal line intersects the graph of the function $f$ in more than one point, the function is one-to-one.

The function $y = x^{3}$ is a one-to-one function.

It is an invertible function.

The function $y = x^{2}$ is NOT a one-to-one function because each $y$ -value occurs twice for different elements of the domain, except at the vertex. It is not an invertible function.

parameter

A number that describes a characteristic of a population (such as the mean or the standard deviation).

parameter of interest (statistics)

A parameter of interest is what your data is focused on.

The thing we want to know about the population.

A number, such as the mean or standard deviation, that describes an entire population. Any numerical quantity that characterizes a given population or some aspect of it. This means the parameter tells us something about the whole population.

parent function

The most basic form of a function. A parent function can be transformed to create a family of functions.

Pascal’s triangle

An array of numbers forming a triangle named after a famous mathematician Blaise Pascal. The top number of the triangle is $1$ , as well as all the numbers on the outer sides. To get any term in the triangle, you find the sum of the two numbers above it. The top number is considered row $0$ of the triangle.

period of a cyclical function

The time it takes for one complete cycle of a cyclical motion to occur. The diagram shows the graph of $y = \sin θ$ . The graph begins at $0$ . At $2 π$ the graph begins to repeat because it has completed one cycle. The period is $2 π$ .

period of rotation

The period of rotation is the amount of time for one complete rotation of the Ferris wheel.

phase shift

For trigonometric functions, a horizontal transformation of a graph is referred to as a phase shift.

point of inflection

A point on a curve where the curve changes from being concave down to concave up or vice versa.

In the normal curve it is one standard deviation away in either direction from the mean.

polar coordinates

A method of representing points in a plane with ordered pairs in the form $(r, θ),$ where $r$ is the distance of the point from the origin and $θ$ is the angle of rotation of the point from the positive $x$ -axis.

polynomial function

A function of the form:

$y = a_{0} x^{n} + a_{1} x^{n - 1} + a_{2} x^{n - 2} + \dots a_{n - 3} x^{3} + a_{n - 2} x^{2} + a_{n - 1} x + a_{n}$

where all of the exponents are positive integers and all of the coefficients $a_{0} . . . a_{n}$ are constants.

population (in statistics)

The group of individuals you want to study in order to answer your research question

population mean

The population mean is an average of a group characteristic or item of interest.

The symbol ‘ $μ$ ' represents the population mean.

population parameter

A population parameter is the actual value of a statistical measure such as the mean or standard deviation for a given population.

population proportion

A population proportion is a fraction of the population that has a certain characteristic. The letter $p$ is used for the population proportion. It can be written as a fraction e.g. $p = \frac{156}{1000}$ or as a decimal $p = 0.156$ .

product

The result of multiplication is a product.

quadrantal angle

An angle in standard position with terminal side on the $x$ -axis or $y$ -axis. Some examples are the angles located at $0 °$ , $90 °$ , $180 °$ , $270 °$ , $360 °$ .

quadratic equations

An equation that can be written in the form $y = a x^{2} + b x + c, where a \geq 1$

Standard form: $f (x) = a x^{2} + b x + c, where a \neq 0$

Example: $x^{2} - 2 x - 8 = 0$

Factored form: $(x - 4) (x + 2) = 0$

Vertex form: ${(x - 1)}^{2} - 9 = 0$

Recursive form: $f (1) = - 9; f (n) = f (n - 1) + (2 n - 3)$

(Note: Recursive forms are only used when the function is discrete.)

quadratic function

quantity

A quantity is an amount, number, or measurement. It answers the question “How much?”

quotient

See division.

radian

A unit of measure for angles. One radian is the angle made at the center of a circle by an arc whose length is equal to the radius of the circle.

The ratio of the length of an intercepted arc to the radius of the circle on which that arc lies.

range of a function

All the resulting $y$ -values obtained after substituting all the possible $x$ -values into a function. All of the possible outputs of a function. The values in the range are also called dependent variables.

rate of change (slope)

A rate that describes how the output of a function changes in relation to the input.

Functions are defined by their rates of change.

In a linear function, if $x$ is the independent variable and $y$ is the dependent variable, the rate of change equals $\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ and is called the slope.

An exponential function has an exponential rate of change.

A quadratic function has a linear rate of change.

A cubic has a quadratic rate of change.

rational function

A function $f (x)$ is called a rational function if and only if it can be written in the form $f (x) = \frac{P (x)}{Q (x)}$ where $P$ and $Q$ are polynomials in $x$ and $Q$ is not the zero polynomial.

reciprocal trigonometric functions

The reciprocal of the sine, cosine, and tangent ratios.

$Cosecant – \csc A = \frac{1}{\sin A} = \frac{hypotenuse}{opposite} = \frac{c}{a}$

$Secant – \sec A = \frac{1}{\cos A} = \frac{hypotenuse}{adjacent} = \frac{c}{b}$

$Cotangent – \cot A = \frac{1}{\tan A} = \frac{adjacent}{opposite} = \frac{b}{a}$

rectangular coordinate system

Also called the Cartesian coordinate system, it’s the two-dimensional plane that allows us to see the shape of a function by graphing.

Each point in the plane is defined by an ordered pair. Order matters! The first number is always the $x$ -coordinate; the second is the $y$ -coordinate.

recursive equation

Also called recursive formula or recursive rule. See examples under arithmetic sequence and geometric sequence, and quadratic equations.

reference triangle

A right triangle that is drawn connecting the terminal ray of an angle in standard position to the $x$ -axis. In the diagram, $△ A C D$ is the reference triangle.

reflection

A reflection is a rigid transformation (isometry). In a reflection, the pre-image and image points are the same distance from the line of reflection; the segment connecting corresponding points is perpendicular to the line of reflection.

The orientation of the image is reversed.

remainder

See division.

remainder theorem for polynomials

When a polynomial f $(x)$ is divided by $x - a$ , the remainder equals $f (a)$ .

Why is this true? The division algorithm can be used to prove the remainder theorem.

restricted domain

Limiting the domain of a function so that its inverse is also a function.

roots: real and imaginary

The solutions of an equation in the form $f (x) = 0$ .

S–X

sample

A part of a population selected to represent the entire population. Sampling is the process of selecting and studying a sample from a population in order to make conjectures about the entire population. A good sample represents the target population.

Types of samples:

simple random sample - one in which every possible sample (of the same size) has an equal chance of being selected from the target population.

systematic sample – A method of choosing a random sample from among a larger population. The process of systematic sampling typically involves first selecting a fixed starting point in the larger population and then obtaining subsequent observations by using a constant interval between samples taken.

cluster sample – With cluster sampling, the researcher divides the population into separate groups, called clusters. Then, a simple random sample of clusters is selected from the population. The researcher conducts his analysis on data from the sampled clusters.

stratified random sample – With stratified sampling, the researcher divides the population into separate groups, called strata. Then, a probability sample (often a simple random sample ) is drawn from each group.

convenience sample – made up of people who are easy to reach.

volunteer sample - made up of people who self-select into the survey.

sample mean

The sample mean is simply the average of all the measurements in the sample. If the sample is random, then the sample mean can be used to estimate the population mean. The symbol for sample mean is $\overset{―}{x}$ ( $x$ bar)

sample proportion

the proportion of people from the sample who fell into a certain group

The symbol is $\hat{p}$ ( $p$ - hat).

sample statistic

A statistic or sample statistic is any quantity computed from values in a sample that is used for a statistical purpose. It’s a piece of information you get from a fraction of a population.

simple random sample

See sample.

simulation

A model of random events, often using technology

One purpose is to test a hypothesis without doing an actual experiment. They are sometimes used because they are cheaper, faster, and less risky than an actual study.

skewed distribution

When most data is to one side leaving the other with a ‘tail’. Data is skewed to side of tail. (if tail is on right side of data, then it is skewed right).

slant asymptote

See asymptote.

slope

A linear function has a constant slope $(m)$ or rate of change. You can count the slope of a line on a graph by counting how much it changes vertically each time you move one unit horizontally. A move down is negative and a move to the left is negative.

If you know two points on the graph, you can use the slope formula. Given two different points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$

$m = \frac{(y_{2} - y_{1})}{(x_{2} - x_{1})}$

$m$ is the symbol for slope.

solids of revolution

A 3-D object formed by spinning a 2-D figure about an axis.

A disk is a slice of the solid of revolution. Each disk’s face is a circle.

A washer is a slice of a hollow solid of revolution. Its face is a circle with a hole in the center.

special right triangles

There are two special right triangles. They are special because they can be solved without using trigonometry.

standard deviation

A number used to tell how measurements for a group are spread out from the average (mean), or expected value. A low standard deviation means that most of the numbers are close to the average. A high standard deviation means that the numbers are more spread out. Symbol for standard deviation $σ$ . (sigma)

standard normal distribution

The standard normal distribution is a normal distribution with a mean of zero and standard deviation of 1. The standard normal distribution is centered at zero and the degree to which a given measurement deviates from the mean is given by the standard deviation.

* normal distributions do not necessarily have the same means and standard deviations

stratified random sample

See sample.

subtraction of polynomials

Subtraction and addition are opposite operations. This is true with polynomials. The diagram shows how the parts of an addition problem and a subtraction problem are related.

subtrahend

See subtraction of polynomials.

sum and difference identities

See trigonometric identities.

symbols for sample statistics and corresponding population parameters

Sample Statistic	Population Parameter	Description
$n$	$N$	number of members of sample or population
$\overset{―}{x}$ “ $x$ -bar”	$μ$ “mu”	mean
$s$	$σ$ “sigma”	standard deviation
$r$	$ρ$ “rho”	coefficient of linear correlation
$\hat{p}$ “ $p$ -hat”	$p$	proportion

systematic sample

See sample.

terminal ray or side

The side of an angle in standard position that is not on the positive $x$ -axis but has an endpoint at the origin or center of rotation.

transformations on a function (rigid)

A shift up, down, left, or right, or a vertical or horizontal reflection on the graph of a function is called a rigid transformation.

Vertical shift

$f (x) + k$

Up when $k > 0$

Down when $k < 0$

Horizontal shift

$f (x + k)$

Left when $k > 0$

Right when $k < 0$

Reflection

$- f (x)$ reflection over the $x$ -axis

$f (- x)$ reflection over the $y$ -axis

A dilation is a nonrigid transformation. It will make the function changes faster or slower depending on the value of $k f (x)$ . If $k > 1$ , it will grow faster and look like it has been stretched. If $0 < k < 1$ , the function will grow more slowly and will appear to be fatter.

treatment group

The treatment group consists of participants who receive the experimental treatment whose effect is being studied.

The control group consists of participants who do not receive the experimental treatment being studied.

trigonometric functions

Right triangle trigonometry can be extended to define trigonometric functions for angles of rotation of any value, including negative values. To do so, a standard position for an angle of rotation is defined: locate the vertex of the angle of rotation at the origin of a Cartesian coordinate system, with the initial ray pointed along the positive $x$ -axis. A counter-clockwise rotation is considered positive while a clockwise rotation is considered negative. With this new definition of the trigonometric functions, trigonometry can be applied to periodic behavior.

trigonometric identities

Statements that are true for all values of $θ$ (theta).

Tangent and Cotangent Identities	$\tan θ = \frac{\sin θ}{\cos θ}$ $\cot θ = \frac{\cos θ}{\sin θ}$
Reciprocal Identities	$\csc θ = \frac{1}{\sin θ}$ $\sin θ = \frac{1}{\csc θ}$ $\sec θ = \frac{1}{\cos θ}$ $\cos θ = \frac{1}{\sec θ}$ $\cot θ = \frac{1}{\tan θ}$ $\tan θ = \frac{1}{\cot θ}$
Pythagorean Identities	$\sin^{2} θ + c o s^{2} θ = 1$ $\sin^{2} θ = 1 - \cos^{2} θ$ $\cos^{2} θ = 1 - \sin^{2} θ$
Even/Odd Formulas	$\sin (- θ) = - \sin θ$ $\cos (- θ) = \cos θ$ $\tan (- θ) = - \tan θ$
Periodic Formulas	If $n$ is an integer, $\sin (θ + 2 π n) = \sin θ$ $\cos (θ + 2 π n) = \cos θ$ $\tan (θ + π n) = \tan θ$
Double Angle Formulas	$\sin (2 θ) = 2 \sin θ \cos θ$ $\cos (2 θ) = \cos^{2} θ - \sin^{2} θ$ $\cos (2 θ) = 2 \cos^{2} θ - 1$ $\cos (2 θ) = 1 - 2 \sin^{2} θ$
Sum and Difference Formulas	$\begin{matrix} \sin (α \pm β) = \\ \sin α \cos β \pm \cos α \sin β \end{matrix}$ $\begin{matrix} \cos (α \pm β) = \\ \cos α \cos β \pm \sin α \sin β \end{matrix}$

If $x$ is an angle in degrees and $b$ is an angle in radians, then:

$\frac{π}{180} = \frac{b}{x} ∴ b = \frac{π x}{180} and x = \frac{180 b}{π}$

Definitions of the Inverse Trigonometric Functions
$y = \sin^{- 1} x$ Domain: $[- 1, 1]$ Range: $[- \frac{π}{2}, \frac{π}{2}]$
$y = \cos^{- 1} x$ Domain: $[- 1, 1]$ Range: $[0, π]$
$y = \tan^{- 1} x$ Domain: $(- \infty, \infty)$ Range: $(- \frac{π}{2}, \frac{π}{2})$

Definitions of the Inverse Trigonometric Functions

$y = \sin^{- 1} x$

Domain: $[- 1, 1]$

Range: $[- \frac{π}{2}, \frac{π}{2}]$

$y = \cos^{- 1} x$

Domain: $[- 1, 1]$

Range: $[0, π]$

$y = \tan^{- 1} x$

Domain: $(- \infty, \infty)$

Range: $(- \frac{π}{2}, \frac{π}{2})$

trinomial

A polynomial with three terms.

unimodal

See mode(s).

unit circle

A circle with radius of one unit. The equation of a unit circle with center $(0, 0)$ is $x^{2} + y^{2} = 1$ .

The unit circle is a useful tool when studying trigonometric functions.

Radian measure is the ratio $\frac{arc length}{radius}$ . On the unit circle $r = 1$ , so the radian measure is the arc length.
Sine $θ$ is the ratio $\frac{y}{r}$ . On the unit circle $r = 1$ , so the sine is the $y$ -coordinate.
Cosine $θ$ is the ratio $\frac{x}{r}$ . On the unit circle $r = 1$ , so the cosine is the $x$ -coordinate.

Example: In the unit circle shown, point $B$ is defined by the coordinates $(\frac{\sqrt{3}}{2}, \frac{1}{2})$ . Since $r = 1$ , $\sin 30 °$ is $\frac{1}{2}$ and $\cos 30 °$ is $\frac{\sqrt{3}}{2}$ . The arc length is $\frac{2 π}{12}$ or $\frac{π}{6}$ .

vertical asymptote

See asymptote.

vertical height

The perpendicular distance from the ground up to a designated position.

vertical shift

See transformations on a function (rigid).

vertical stretch

See transformations on a function (non-rigid).

volunteer sample

See sample.

washer

A washer is a slice of a hollow solid of revolution. Its face is a circle with a hole in the center.

x-intercept

The point(s) where a line or a curve cross the $x$ -axis. The $y$ -value of the point will be $0$ . A non-horizontal line will only cross the $x$ -axis once. A curve could cross the $x$ -axis several times.

Y–Z

z-score

The number of standard deviations that a given $x$ -value lies from the mean in a normal distribution. The formula for transforming a data point from any normal distribution to a standard normal distribution:

$z -score = \frac{data point - mean}{standard deviation}$

zeros (of a function)

The values of the independent variable (x-values) that make the corresponding values of the function ( $f (x)$ -values) equal to zero. Real zeros correspond to $x$ -intercepts of the graph of a function.