Table of Contents
Trig Magic Hexagon
A trig magic hexagon is a hexagon-shaped array of trigonometric functions that are related to each other. The six trigonometric functions that are typically used in a magic hexagon are sine, cosine, tangent, cosecant, secant, and cotangent. Each of the six functions is related to two of the others, and these relationships are shown in the hexagon. The relationships are as follows:
sine = cosine
tangent = secant
cosecant = cotangent
cosecant = secant
cosine = tangent
Normal Magical Hexagons
Normal magical hexagons are hexagons that are not associated with any specific magical properties. They are typically used for decorative purposes or in spells that do not require any specific magical properties.
Trigonometry Hexagon
A hexagon is a six-sided polygon. It has six angles and six sides. The angles of a hexagon can be found by using the trigonometric functions sine, cosine, and tangent. The length of the sides of a hexagon can also be found using the trigonometric functions sine, cosine, and tangent.
Building the Trig Hexagon Identities
The trigonometric functions are periodic, which means that they repeat over and over again. The period of a function is the amount of time it takes for the function to repeat.
The trigonometric functions have a period of 360 degrees. This means that they repeat every 360 degrees.
The trigonometric functions can be used to find the exact value of a trigonometric function for any angle. However, not all angles can be written in standard form.
Standard form is when an angle is written in terms of its radian measure. Radian measure is the number of degrees in a circle divided by the number of radians in a circle. There are 2π radians in a circle.
There are a few angles that can be written in standard form, and these are the angles that we will be working with in this section.
The six angles that can be written in standard form are: 0°, 30°, 45°, 60°, 90°, and 180°.
The Trig Hexagon
The trigonometric functions can be used to find the exact value of a trigonometric function for any angle. However, not all angles can be written in standard form.
Standard form is when an angle is written in terms of its radian measure. Radian measure is the number of degrees in a circle divided by the number of radians in a circle. There are 2π rad
Pythagorean Identities
There are a few Pythagorean Identities that are worth memorizing.
1. The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
2. The cosine of an angle is equal to the ratio of the length of the adjacent side to the length of the hypotenuse.
3. The sine of an angle is equal to the ratio of the length of the opposite side to the length of the hypotenuse.
Deriving the Pythagorean Identities
The Pythagorean Identities are a set of three mathematical identities that are named after the ancient Greek mathematician Pythagoras. These identities relate the Pythagorean theorem to other mathematical concepts.
The Pythagorean theorem states that in a right angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. The Pythagorean Identities allow us to relate this theorem to other mathematical concepts.
The first Pythagorean identity states that the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides, plus the square of the length of the hypotenuse.
The second Pythagorean identity states that the square of the length of the hypotenuse is equal to the product of the squares of the other two sides, minus the square of the length of the hypotenuse.
The third Pythagorean identity states that the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides, minus the square of the length of the longest side.
Even and Odd Identities
An odd identity is an equation that states that the sum of two odd numbers is an odd number. An even identity is an equation that states that the sum of two even numbers is an even number.
The sum of two odd numbers is always an odd number.
The sum of two even numbers is always an even number.
