A beginner’s overview of Trigonometry
Some students talk about Trigonometry. What is it all about?
It is a type of Math. It is to do with Ratios and Geometry. Everything in this world has a geometry. From flowers to animals to humans, each of us fall into a distinct shape and ratios.. One might be a big tall man and another a short person, but the ratio of hands, legs and head is identical right..
The same idea has been extended into triangles.. Trigonometry a Greek word meaning ‘measurements’ with ‘triangles’ comes from the Sanskrit word ‘trikonamati’
Exploring the Right Angled Triangles
Think of a one year old baby.. He is sleeping on the bed with tummy down.. Then he lifts his chest and head like snake pose.. He then is on his fours.. and slowly stands up on two legs. His length was same through out but slowly he was getting upwards..
Let’s make a journey into the land of Right Angled Triangles.. See what they have to tell.
There are different kinds of them..
The sleeping beauty to begin with.. totally horizontal and of pigmy height
At 15 degree slowly waking up with a small height
Growing taller and thinner at degree 30..45..60..75
And totally tall and awake at 90 degree..
From Horizontal and a tiny height, it has slowly become Vertical and Tall.. Swimming vs Running, Fish vs Giraffe, Horizontal Sun in the evening vs Vertical Sun in the Noon.. Truck vs Aeroplane
What can we deduce from this? When the angle increases, the height also increases.. This fact was termed as ‘Sine’.. It is the ratio of height and hypotenuse.. It was initially 0 due to small height and slowly it became 1 as both the lengths became identical.
You could try this.. Close both your palms in the 3.15 clock position.. With a string, tie a pencil to the edge of your left hand little finger.. Slowly rotate your left palm anticlockwise till it becomes vertical and shows 3.00 position.. Did you observe the continuous set of RightAngledTriangles and the height gradually increasing?
So Sine is related to height. What are the other ratios?
Good question.. Think about the other combinations.. Since there are 3 sides a, b and c, we can have 6 ratios.. a/b, c/b, a/c and their reciprocals.. These ratios are given names as sine, cosine, tangent and their reciprocals cosecant, secant and cotangent..
So many ratios to understand..
If you have understood Sine, you have pretty much understood everything.. Remember, it is a Right Angled Triangle.. If you know two sides, you could get the third Side.. Pythagoras Theorem comes to your rescue.
An interesting video to show the Sine and Tangent function. It is a circle of radius and hypotenuse of length 1 unit. Watch the height of the triangle increasing. That corresponds to the Sine of the angle. Watch the green line which is just touching the circle at the exterior. The tangent is shooting to infinity as the angle becomes perpendicular.
Ahh.. These waves keep repeating..
Why these cryptic names? Is there a meaning to it?
Well.. During the Gupta period, there was a famous mathematician called Aryabhatta in India. He called the Sine function as ‘Ardha Jya’ or ‘Jeeva’ meaning ‘half chord’. Remember the circle figure and the height. It resembled chord of a bow The Arabs wrongly translated ‘Jeeva’ to ‘Jaib’ meaning pocket. In Latin, it was translated to ‘Sinus’ meaning pocket or cavity (like the nasal cavities in the skull)..
The Cosine function is just Complementary to Sine.. So called as Cosine..
What can we do with Trigonometry?
Imagine you placed a ladder of length 10m against a wall making an angle of 60 degrees. Can we compute the height of the wall?
Well, Sine of 60 is ratio of height AB to the ladder length AC.. Using a trigonometry table, you could get the Sine ratio for any angle. And Bingo.. Ladder length times Sine value is your Wall length.
Ahh. There is a book having ratios for different angles.. How did they make that.. Did they actually construct different triangles and measure..?
Well, the ancient mathematicians did a good job of quick approximating. They also had long long expressions for accurately computing these values.
Ancient Babylonians, Indians, Chinese, Arabs needed Trigonometry for calculating astronomical distances and in Constructions.
OHH.. It is an age old Practical Math. Where else is it used in our times ?
Let’s discuss BasketBall. When you throw the ball in the air, you do it with an angle unlike striking a carrom coin. There is a vertical push making it go up and a forward push making it go front simultaneously.. We have seen before that the Sine of the angle always determines the height portion..
The Radio waves of the Cell tower and the AC current (alternating current) travel as Sine waves.. Remember short height progresses to long height.. like a wave it keeps repeating…
In the words of Jain 108 on the Sine and Cosine as Artistic Waves.
“Don’t let the scary, frightening Trigonometry, that was thrust upon you in your early high-school days, haunt you anymore. There is a shift happening, through the perception of Art. Just know that these are beautiful waves describing reality at intersecting 90 degree angles.
Sine and Cosine are orthogonal, 2-dimensional projections of the 3-dimensional Unit Helix, which is really the Circle rotated through Space-Time.
The Helix is the Spiral, the DNA, and the Unit Cube is 1x1x1.
That’s as simple as it gets. With new eyes, and new perspective, the pioneering and geometric Maths of Euler’s formula can be an ever-evolving and fascinating vibrational Language of Light.”