Sine, Cosine and Tangent - Simply heights and lengths.

Sines, Cosines and Tangents are simply different, complicated words for people who learns maths. Any math student will addition to be very useful and subtraction to be equally useful and at time multiplication is needed. while most may wonder why do we need to study trigonometry.

Many people who learn pythagoras theorem will appreciate to calculate the unknown one side length of the triangle when 2 of the other sides are known with the formula c^2 = a^2 +b^2.

Trigonometry: Tri+gono+metry  = Three+Angle+Measure

From my point of view 1st 2D primitive and the only primitive 2D shape is traingle with least no. of sides. All other 2D shapes have more than 3 sides. we cannot form 2D shape with 2 lines alone, we require 3 lines.

Based on sides triangles can be equilateral, isosceles and scalene.

Based on angles triangles can be right angled, acute angled and obtuse angled.

Equiangular triangle - equilateral by default whose angle is 60 degrees. Isoceles triangle - right angled by default.

It is also interesting to note that triangles have 4 centres - Centroid, Circumcenter, Incenter, and Orthocenter whereas circle has one and ellipsis has 2.

Out of these different kinds of triangles trigonometry uses only the right angle triangle which need not be isosceles triangle. Because that is the only triangle that fits in a semicircle.

Direct use of trigonometry

Trigonometry is useful to calculate heights of mountains building etc with trail and error method. With one angle and one side being measured.

Indirect use of trigonometry

Sine and Cosine wave linked with exponential wave forms are useful waveforms to denote currents, growth and potentials. It is also used in signal processing and analysis, waveform analysis, to create filters and osciloscopes.

Tangential waves are used in shear testing like echo and non destructive testing.

Sines and Cosines are simply height and lengths:

Given a unit circle (1 unit length) and a line from centre touching the circumference and a horizontal diameter line.

Sine is nothing but the height of the line from horizontal line. Cosine is the nothing but the length of the  shadow of the line falling on the horizontal line.

while we can measure the height and the length of the shadow with a scale, they are always expressed as ratios with respect to angles, why?

For example 

sine (theta) i.e height projection of the line = opposite projection length (height) / hypotenuse (1 unit length)

cosine (theta) i.e height projection of the line = adjacent projection length / hypotenuse (1 unit length)

From the ratio description we can see, it can be applied for any unit length circle and any right angled triangle.

Hypotenuse is the radius of the circle, in this case it refers to the side of the triangle which is under tension. Hypotenuse = under+tension. Earlier days a thread under tension was used to draw a circle, I hope it gave the name hypotenuse.

Sinus means curve or bay, Sine is nothing but the bay and cosine its sister or companion. if there is radiation from the angle it will stop at the bay or the opposite.

Cosine is actually called chord which means only a string chord. Today we say chord as a line whose ends touches the circle.

Tangent is the line touching the circle Tan(theta) = sin(theta)/cos(theta) it is actually the ratio of height by length, indicates a measure of how steeply the height is with respect to its shadow.