Glossary

An acute angle is a type of angle that measures greater than 0 degrees but less than 90 degrees, meaning that it is smaller than a right angle.

In trigonometry, the adjacent side is the side of a right triangle that is adjacent, or next to, a given angle. It is the side that forms one of the two acute angles in the right triangle, along with the hypotenuse and the opposite side. The adjacent side is always the side that is adjacent to the angle for which we are calculating a trigonometric function such as cosine or tangent

Alternate interior angles are a pair of angles that are formed by a transversal intersecting two parallel lines, located on opposite sides of the transversal and in between the two parallel lines, and are congruent, meaning that they have the same measure.

The altitude of an equilateral triangle divides it into two congruent right triangles.

The elevation angle is the angle between the horizontal plane and the line of sight from an observer to an object or point in space.

In geometry, arc length is the length of a portion of the circumference of a circle, measured along the arc. It is denoted by “s” and can be calculated using the formula [latex]s = r theta[/latex] , where “r” is the radius of the circle and “θ” is the central angle (in radians) that subtends the arc.

A base angle of a shape is an angle within a shape that contains the base of the shape as one of its sides.

A cardioid is defined in the polar coordinate system by the equation:

\[ r = a * (1 + \cos( \theta)) \]

In geometry, a circle is a two-dimensional shape that consists of all points that are a fixed distance (called the radius) from a given point in a plane.

In geometry, the circumference of a circle is the distance around its outer perimeter or boundary. It can be calculated using the formula [latex]C = 2 \pi r[/latex]

Two angles are considered complementary if the sum of their measures is exactly 90 degrees.

Two triangles are congruent if they have exactly the same size and shape.

The converse of a statement is a proposition obtained by reversing its terms. For example, the converse of the statement “If it is raining, then the streets are wet” is “If the streets are wet, then it is raining.” Note that the converse may or may not be true, even if the original statement is true.

Corresponding angles are a pair of angles that are formed by a transversal intersecting two parallel lines, located in corresponding positions in relation to the transversal, and are congruent, meaning that they have the same measure.

Cosine is a trigonometric function equal to the ratio of the length of the adjacent side a given angle in a right triangle to the length of the hypotenuse, expressed as \[ cos ( theta = \frac{\text{opposite}}{\text{hypotenuse}} \].

A decimal approximation for a trigonometric function is an estimation of its value expressed as a finite string of decimal digits, usually rounded to a certain number of decimal places, and obtained through the use of a calculator or other computational tool.

Discriminant \[ (D) = b^2 - 4ac \]

If the discriminant is greater than zero (D > 0), then the quadratic equation has two distinct real solutions. This means the graph of the equation intersects the x-axis at two different points.

If the discriminant is equal to zero (D = 0), then the quadratic equation has one real solution.
If the discriminant is less than zero (D < 0), then the quadratic equation has no real solutions.

An equilateral triangle is a type of triangle in which all three sides are equal in length, and all three angles are also equal, measuring 60 degrees each.

An exact value for a trigonometric function is a value that can be represented as a finite combination of integers, radicals, and/or known mathematical constants without the use of a calculator or approximations.

Extraction of roots refers to the process of finding the value of a root (usually square, cube, or higher) of a number or expression, either by using a numerical method or by solving an equation.

In a 30°-60°-90° right triangle, the leg opposite the 30° angle is half the length of the hypotenuse.

In mathematics, an identity is an equation that is true for all values of the variables involved. In other words, it is a statement that holds true regardless of the values of the variables, and does not need to be proved or solved for a specific set of values. Examples of identity sin²θ + cos²θ = 1.

An inequality is a mathematical statement that compares two values, expressions, or quantities using an inequality symbol (such as “>” for greater than, “>=” for greater than or equal to), indicating that one value is not equal to the other.

An angle is in standard position if its vertex is located at the origin and one ray is on the positive x-axis. The ray on the x-axis is called the initial side and the other ray is called the terminal side.

The inverse cosine (arccosine) of a value is the angle whose sine is equal to that value.

The inverse sine (arcsine) of a value is the angle whose sine is equal to that value.

The inverse tangent (arctangent) of a value is the angle whose sine is equal to that value.

An irrational number is a real number that cannot be expressed as a ratio of two integers, and its decimal expansion is non-repeating and non-terminating.

An isosceles triangle is a type of triangle that has two sides of equal length and two equal angles opposite those sides.

In a 30°-60°-90° right triangle, the leg opposite the 30° angle is half the length of the hypotenuse.

[latex]r^2=a^2 cos 2theta[/latex]

[latex]r^2=a^2 sin 2theta[/latex]

A lemniscate graph is a type of mathematical curve which resembles a figure-eight or an infinity symbol [latex]\infty[/latex]

A limaçon is a curve that can take on various forms, from loops to dimples to heart-like shapes, depending on the values of its parameters
[latex]r=apm bsin theta~[/latex] or [latex]~r=apm bcos theta[/latex]

The magnitude of a vector refers to its size or length and represents the scalar value associated with the vector. It quantifies the extent or amount of the vector without considering its direction. The magnitude of a vector is always a non-negative value.

The minute is a unit of time measurement that is equal to 60 seconds, or 1/60th of an hour

An oblique triangle is a triangle that does not have a right angle. In an oblique triangle, all three angles are acute (less than 90 degrees) or one angle is obtuse (greater than 90 degrees). Oblique triangles are sometimes also referred to as non-right triangles.

An obtuse angle is a type of angle that measures greater than 90 degrees but less than 180 degrees, meaning that it is larger than a right angle.

Parallax is the apparent shift or difference in the position of an object when viewed from different positions. In other words, it is the displacement or difference in the apparent position of an object seen along two different lines of sight.

A parallelogram is a quadrilateral with two pairs of parallel sides, where opposite sides are congruent (have the same length) and opposite angles are also congruent (have the same measure).

Two triangles are similar if they have the same shape but not necessarily the same size. The corresponding angles are equal, and the corresponding sides are proportional.

The Pythagorean theorem states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In mathematical notation, it can be expressed as:

[ c^2 = a^2 + b^2 ]

where [latex]ax[/latex] and [latex]b[/latex] are the lengths of the two legs of the right triangle, and [latex]c[/latex] is the length of the hypotenuse. This theorem is named after the ancient Greek mathematician Pythagoras, who was the first to prove its validity.

A quadrantal angle is an angle in the standard position and has a measure which is a multiple of [latex]90^o[/latex] or [latex]\frac{\pi}{2}[/latex] radians. It is an angle in standard position whose terminal ray lies along one of the axes.

A polynomial equation with degree 2 is called quadratic. The graph of a quadratic is called a parabola.

It is simply written in the form \[ ax^2+bx + c = 0 \], where [latex]a \neq 0[/latex]

\[ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \]

A quadrilateral is a four-sided polygon with straight sides and four vertices.

In geometry, the radius of a circle is the distance between the center of the circle and any point on its circumference. It is usually denoted by the letter “r”.

A rational number is any number that can be expressed as a ratio of two integers, such as [latex]3/5[/latex].

A repeating decimal is a decimal number that has a block of one or more digits that repeat infinitely after the decimal point, such as 0.333..., 0.666..., or 0.142857142857... (where the block 142857 repeats).

A right angle is a type of angle that measures exactly 90 degrees. It is formed when two lines or line segments intersect each other perpendicularly, meaning that they meet at a 90-degree angle.

A right triangle is a type of triangle that has one interior angle measuring exactly 90 degrees, or a “right angle.”

\[ r = a \sin {n\theta} \] or
\[ r = a \cos {n\theta} \]

In mathematics, a scalar is a quantity that only has magnitude and does not have a specific direction associated with it. Scalars are used to represent physical quantities such as mass, temperature, time, distance, and speed.

Unlike vectors, which have both magnitude and direction, scalars can be completely described by a single numerical value or a scalar quantity. Scalars are often represented by regular non-boldface letters or lowercase Greek letters.

The second is a unit of time measurement that is equal to 1/60th of a minute or 1/3,600th of an hour.

Two triangles are similar if they have the same shape but not necessarily the same size.

Sine is a trigonometric function equal to the ratio of the length of the side opposite a given angle in a right triangle to the length of the hypotenuse, expressed as \[ \sin ( \theta )= \frac{\text{opposite} }{\text{hypotenuse}}. \]

If we know three parts of a right triangle, including one of the sides, we can use trigonometry to find all the other unknown parts. This is called solving the triangle.

Special angles in trigonometry are angles whose sine, cosine, and tangent values can be calculated exactly without the use of a calculator.

An angle is in standard position if its vertex is located at the origin and one ray is on the positive x-axis. The ray on the x-axis is called the initial side and the other ray is called the terminal side.

A straight angle is a type of angle that measures exactly 180 degrees. It is formed when two opposite rays or line segments extend from the same point and create a straight line.

Two angles are considered supplementary if the sum of their measures is exactly 180 degrees.

Tangent is a trigonometric function that describes the relationship between the sides of a right triangle. It is equal to the ratio of the length of the side opposite a given angle to the length of the adjacent side, expressed as tan(θ) = opposite/adjacent. In other words, the tangent of an angle in a right triangle is equal to the length of the side opposite the angle divided by the length of the adjacent side.

An angle is in standard position if its vertex is located at the origin and one ray is on the positive x-axis. The ray on the x-axis is called the initial side and the other ray is called the terminal side.

A transversal is a line that intersects two or more other lines or line segments at different points, forming eight angles, four of which are congruent pairs of vertical angles.

The triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the third side.

The unit circle is a circle with a radius of one centered at the origin [latex](0,0)[/latex] of a coordinate plane, often used in trigonometry and geometry as a reference circle. It has equations [latex]x^2 + y^2 = 1[/latex].

In mathematics, a vector is a mathematical object that represents both magnitude and direction. It is commonly used to describe quantities such as displacement, velocity, force, or any other physical or mathematical entity that has both magnitude and direction.
Vectors can also be represented geometrically as directed line segments, where the length of the segment represents the magnitude of the vector, and the direction of the segment represents the direction of the vector.

Vertex angle is defined as the angle formed by two lines or rays that intersect at a point. These two rays make the sides of the angle. In other words, the angle associated within a given vertex is called vertex angle.

Vertical angles are a pair of non-adjacent angles that are opposite each other, formed by the intersection of two lines or line segments, and are always congruent, meaning that they have the same measure.