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A detailed tutorial on how to identify disconnected graphs. Step by step tutorial including several examples of disconnected graphs for reference.

A disconnected graph is a graph where not every single vertex is connected to all other vertices. Typically, graphs will have paths from all vertices, but if there is not a direct path from each and every vertex, then it is considered to be a disconnected graph. Some common shapes that are seen that are disconnected graphs are stars, rectangles, and hexagons. The opposite of a disconnected graph is a connected graph.

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A detailed tutorial on how to identify connected graphs. Step by step tutorial including several examples of connected graphs for reference.

A connected graph is a graph where every single vertex is connected to every other vertex. This does not mean to simply have a clear path from one vertex to another – it means there needs to be a direct path, or an edge, between two vertices. A triangle is a commonly seen shape that is a connected graph. The opposite of a connected graph is a disconnected graph.

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A detailed tutorial on what composite numbers are. Step by step tutorial including several examples of composite numbers and their definition for reference.

A composite number is the opposite of a prime number. Some people say they are any number that is not prime, but that is not exactly accurate – negative numbers are not prime (even negative prime numbers), and a composite number is not a negative number, it is a positive number. A composite number is any positive integer that has more divisors than itself and one – which are the only two numbers a prime number can be divided by.

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A detailed tutorial on how to find the reciprocal of a number. Step by step tutorial including several examples of reciprocals for reference.

A reciprocal is a way of saying the opposite of a number, although it is not a true opposite. A true opposite of a negative number would be a positive number, and a true opposite of a positive number would be a negative number – that is why there are such things as opposite reciprocals. A more accurate name for a recirpocal would be the reverse of a number. In a fraction, the reciprocal of a number is when the numerator and the denominator are flipped. This also works for whole numbers, because you can think of the number as a numerator with denominator one.

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A detailed tutorial on how to find the opposite and adjacent sides of a triangle. Step by step tutorial including several examples of finding the opposite and adjacent sides of a triangle for reference.

When using SOHCAHTOA, you will often see something such as “find the opposite side” or “find the adjacent side.” Unlike the hypotenuse, the opposite and adjacent sides change depending on what angle you are working with. The right angle is found opposite the hypotenuse and you will never be working it. Tip your triangle so that your right angle is balanced across the bottom and left, and your hypotenuse crosses the right. You will be working with the angles on the top and on the bottom right. The adjacent side is one of the sides that forms your angle – one of which is the hypotenuse, so it is the other side. And to find the opposite side, draw a straight line from your angle. The line it crosses should be the one directly across from your angle, and it is the opposite side.

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A detailed tutorial on the property of quasitransitive relations. Step by step tutorial including several examples of quasitransitive relations for reference.

A quasitransitive relation can be mathematically defined as for all x, y, and z belonging to A, if x R y, y R z, ~(y R x), and ~(z R y), then x R z and ~(z R x). In this statement, A is a set, and R is a relation of that set. A quasitransitive relation is considered to be a weak version of a transitive relation. If the relation also happens to be asymmetric, then it is considered transitive.

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A detailed tutorial on identifying coterminal angles. Step by step tutorial including several examples of how to identify coterminal angles for reference.

Coterminal angles are opposite angles that when put together share a terminal side, or common side, and therefore create a circle. One of the angles is positive, and the other angle is negative – a negative angle is one that is formed from the opposite side and using the second scale on a protractor. The absolute value of the first angle plus the absolute value of the second angle must add up to 360 degrees in order for them to be coterminal angles.

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A detailed tutorial on complements in set theory. Step by step tutorial including several examples of complements in set theory for reference.

In set theory, a complement is the opposite of something. It works a little like negation, in that the complement of a set is everything but that set. The way to find this is to subtract the set from its universe, which is a larger set that the set you are taking a complement of belongs to. You can think of your set as a subset of the universe.

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A detailed tutorial on how to translate and symbolize negation. Step by step tutorial including several examples of how to solve problems with negation for reference.

Negation is a term used in discrete math that refers to the negation or opposite of a statement, which is represented by the ~ symbol before the statement or letter representing the statement. Negation will change a true statement into a false statement, or a false statement into a true statement. When used in statements instead of letters, the implication of a statement is changed instead of the statement itself.

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Description

A detailed tutorial on the solving of SOHCAHTOA. Step by step tutorial including several examples of how to solve SOHCAHTOA problems for reference.

Overview

SOHCAHTOA, often spaced out to spell SOH-CAH-TOA, stands for Sine = Opposite/Hypontenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. You use it with an angle to help solve for the sine, cosine, or tangent of that angle. What Opposite, Adjacent, and Hyptonuse stand for are the sides of a triangle – the side exactly opposite your angle, the hypotenuse, and the third non-hypotenuse side that is next to your angle. Because of this, SOHCAHTOA can only be used with a right triangle. The values for opposite, adjacent, and hypotenuse are the length of the side of the triangle it stands for. It is not necessary to know the measure of the angle before using SOHCAHTOA.