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trignometry




Trigonometric equations

Equation involving trigonometric function defined above are call trigonemtric equation. Since we know that value of sin and cos function repeat after 2π interval and value of tan function repeat after π interval.So there will be infinite solution for these equations.We define two forms of solutions here

Principle Solution: The solution in the rangle 0 ≤ x ≤ 2π

General Solution: The expression involving integer n which gives all solutions of a trigonometric equation is called the general solution.

We shall use 'Z' to denote the set of integers.

Some Important points in that regard

(a)
sin x = 0 implies x = nπ, where n is any integer
cos x = 0 implies x = (2n + 1)(π/2)
(b)
sinx =siny then x=nπ+(-1)ny where n is any integer
cosx=cosy then x=2nπ+y or x=2nπ- y where n is any integer
tanx=tany then x=nπ+y or x=nπ- y where n is any integer

Some basics Tips to solve the trigonometry questions

1) Always try to bring the multiple angles to single angles using basic formla.Make sure all your angles are the same. Using sin(2X) and sinX is difficult, but if you use sin2X = 2sin(x)cos(x), that leaves sin(x) and cos(x), and now all your functions match.
The same goes for addition and subtraction: don't try working with sin(X+Y) and sinX. Instead, use sin(X+Y) = sin(x)cos(y)+cos(x)sin(y) so that all the angles match
2) Converting to sin and cos all the items in the problem using basic formula. I have mentioned sin and cos as they are easy to solve.You can use any other also.
3) Check all the angles for sums and differences and use the appropriate identities to remove them.
4) Use phythagorean identifies to simplfy the equations
5) Practice and Practice. You will soon start figuring out the equation and there symmetry to resolve them fast



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