Trigonometry

Just to make it easier to see (and write), let's call sin(x) 'S' for a while.

2 S2 = 1 - S

Add S to each side ==> ( 2 S2 + S = 1 )

Subtract 1 from each side:

2 S2 + S - 1 = 0

This is just a plain old quadratic equation, which is easy to factor and solve.

(2S - 1) (S + 1) = 0

2S - 1 = 0

2S = 1

S = 0.5

S + 1 = 0

S = -1

Those are the two solutions for 'S'.

Now it's time to recall that 'S' is really sin(x).

sin(x) = 0.5

x = 30° and 150°

sin(x) = -1

x = 270°

I want to know the basic enhancement of trigonometry I am a college student, taking course of Bachelor of Science in Mathematics

If you are at college studying mathmatics at such a level your previous schooling should have taught you trigonometry. If you do not understand the relationships then you have insufficient grounding for the course on which you are on. I would suggest further study of the basics before you advance. This will save your time and that of your tutors.

There are few uses for trigonometry is schools just as there are few uses for languages or history or geography or, in fact, most subjects. There are three main objectives in teaching most subjects at school:

• one is to give you a taster of that subject. If you had never come across it you would never know whether or not you wanted to study it.
• a second is to provide you with enough of the basics so that, if you do choose to study a related subject at a later stage, you would not have to start from scratch. For example, if they did not teach you any mathematics, you could never be a scientist or, alternatively, your first year of studying proper science, you would have to learn basic addition!
  
• the third aim, is to provide you with enough basic skills to be able to perform effectively as a productive citizen in society. Despite being a mathematician, and someone who loved trig at school, for most people, this argument does not work.

Iteration, in mathematics, is a process of solving an equation numerically. Iteration involves making an estimate for a solution to a problem. That, estimate is used as an input to make another estimate. Then that estimate as the input to get yet another estimate.

With a reasonable first estimate and an appropriate iteration procedure, the answers will converge towards the true solution. However, be aware that in some cases the process can diverge (or oscillate).