On Monday, the Onion reported that the “Nation’s math teachers introduce 27 new trig functions”. It’s a funny read. The gamsin, negtan, and cosvnx from the Onion article are fictional, but the piece has a kernel of truth: there are 10 secret trig functions you’ve never heard of, and they have delightful names like ‘haversine’ and ‘exsecant’.
Trigonometry and its weird names have baffled me for ages. I decided to put an end to this madness by trying to understand them ;P
Here is my interpretation on what thy mean. You can find a detailed etymology here.
Sine ( meaning ~ bowstring in sanskrit )is ratio of the length of the bowstring with the diameter of the circle that the bow makes.
Sin(theta) because this ratio is solely dependent on the angle ‘theta’. Even the slightest change in theta messes up this ratio.
So, when theta = 90 (or 2*theta = 180) , we can see that x = d. This is why sin(90)= 1.
‘Compliment sine’ ( ratio in the perpendicular direction of sine )
Cosine is the ratio of the distance between the bowstring and the center of the bow circle to the diameter of the circle.
So, when theta = 0 , we can see that y = d. This is why cos(0)= 1.
Sine2 + cosine 2 = 1 ?
Well.. thats because of our old friend Pythagoras
Have a great day!
Imagine you are the center of the circle and shoot an arrow through
the point (x,y). Your friend is on the circumference of the circle and
starts running perpendicular to the circle.
Tangent is the distance that your friend needs to cover in order to catch that arrow.
Heres why the tangent is sin/cos
Why so many functions ?
It is important to realize that the reason why we have so many trigonometry functions: Trigonometric functions are merely ratios of distances.
Before the days of calculators, these ratios were manually calculated, and as a result, many functions were introduced and made into tables to ease the computation.
By virtue of everyday usage, the fact that (-1) x (-1) = 1 has been engraved onto our heads. But, only recently did I actually sit down to explore why, in general negative times negative yields a positive number !
Let’s play a game called “continue the pattern”. You would be surprised, how intuitive the results are:
2 x 3 = 6
2 x 2 = 4
2 x 1 = 2
2 x 0 = 0
2 x (-1) = ?? (Answer : -2 )
2 x (-2 ) = ?? (Answer : -4 )
2 x ( -3) = ?? (Answer : -6 )
The number on the right-hand side keeps decreasing by 2 !
Therefore positive x negative = negative.
2 x -3 = -6
1 x -3 = -3
0 x -3 = 0
-1 x -3 = ?? (Answer : 3)
-2 x -3 = ?? (Answer : 6)
The number on the right-hand side keeps increasing by 3.
Therefore negative x negative = positive.
Pretty Awesome, right? But, let’s up the ante and compliment our intuition.
The Number Line Approach.
Imagine a number line on which you walk. Multiplying x*y is taking x steps, each of size y.
Negative steps require you to face the negative end of the line before you start walking and negative step sizes are backward (i.e., heel first) steps.
So, -x*-y means to stand on zero, face in the negative direction, and then take x backward steps, each of size y.
Ergo, -1 x -1 means to stand on 0, face in the negative direction, and then take 1 backward step. This lands us smack right on +1 !
The Complex Numbers Approach.
The “i” in a complex number is an Instruction! An instruction to turn 90 degrees in the counterclockwise direction. Then i * i would be an instruction to turn 180 degrees. ( i x i = -1 ). where i = √-1
Similarly ( -1 ) x i x i = (- 1 ) x ( -1 )= 1. A complete revolution renders you back to +1.
We can snug in conveniently with the knowledge of complex numbers. But, complex numbers were established only in the 16th century and the fact that negative time negative yields a positive number was well established before that.
Hope you enjoyed the post and Pardon me if you found this to be rudimentary for your taste. This post was inspired by Joseph H. Silverman’s Book – A friendly Introduction to Number Theory. If you are passionate about numbers or math, in general it is a must read.
There are several other arithmetic methods that prove the same, if you are interested feel free to explore.