A Tangent on Trigonometry

Isaac Carstensen. 15 October, 2016.

Many people are aware of the trigonometry functions of Sine, Cosine and Tangent (or sin, cos and tan), and most mathematicians are aware of their functions and how useful they are. However, not many people are aware of their origin, how its developed or what the different trigonometry functions are.

If you were not aware, the different trigonometry functions are used to find different sides and angles in a right-angled triangle. Sineθ = Opposite/Hypotenuse, Cosineθ = Adjacent/Hypotenuse, Tangentθ = Opposite/Adjacent, where θ equals the appropriate angle and Opposite, Adjacent and Hypotenuse represent a side relative to the angle.

Summation defines the different functions of trigonometry, each one being a variation of another.

The formula below makes use of factorials. Factorials are represented by the symbol “!”. When you apply factorials to a number, the number is multiplied the value by every other positive integer before it, so for example, 3!=1×2 ×3. So 3! = 6.

Both the functions of Sine and Cosine use the same method but with different numbers, however Tangent is quite different. Tangent occurs by doing the calculation of Sine/Cosine, meaning it uses features of both the original functions.

There’s a 4th trigonometry function which occurs using a value called “i”. This is the theoretical existence of the square root of -1.i is not different from normal numbers, except for when it’s multiplied by itself. For example i²= -1, i³= -i, i⁴=1, i⁵=i, i⁶= -1 etc.

If you were to say that the value of x in Cosine x was i, then each value that i was applied to would reoccur between a positive and negative value. If you were to do the same process but with Sine instead of Cosine, then every number i was applied to would still be in terms of i, due to the fact that the indices don’t remove the i because the indices are odd.

This function is given by the calculation e^ix. “e” is a number similar to pi (π) because it is an irrational number, meaning its value continues without ever repeating or stopping. The 4th function states that e^ix = cos x + isin x, which is fascinating because it uses an imaginary number to give a real value. 
e^ix can be proved by using something known as the Taylor Expansion which uses the equation e^x =1 +x/1!+ x²/2!+x³/3!+… with this equation, if you replace x with an imaginary number (let’s call it z) it creates the calculation of 1+z+z²/2!+z³/3!+etc.

If you replace the value of z with ix, you get the same calculation, except in terms of x and i instead of z. This causes some of the values to change from positive to negative due to the fact when i is to the power of certain numbers, it becomes a negative value.

All these calculations and formulas prove how useful trigonometry is in mathematics and how it can span to all areas of mathematics and how significant it actually is.


Euler’s Formula for Complex Numbers

Further Reading:

Institute of Mathematics: A Beautiful Equation