Solving Basic Equations

It is important to write down all of the solutions in the interval that you are given

Example

Solve sin(x - p/2) = ½ for 0 < x < 2p

We take arcsin [arcsin means sin^-1] of both sides to get:

x - p/2 = arcsin(½)

x - p/2 = -7p/6 , p/6, 5p/6, 13p/6 ...

We want all of the solutions for x between 0 and 2p. You must be careful, because when you take p/2 to the right hand side, the solutions are each going to have increased by p/2 and so some solutions might have entered or exited the range that you want them to be in.

x = 2p/3, 4p/3

Example

Solve 2cos²x + 3sinx = 3, giving your answer in radians for 0< x <p.
2cos²x + 3sinx - 3 = 0
We need to get everything in terms of sinx or everything in terms of cosx. Since we know that cos²x = 1 - sin²x:
2(1 - sin²x) + 3sinx - 3 = 0
2 - 2sin²x + 3sinx - 3 = 0
-2sin²x + 3sinx - 1 = 0
2sin²x - 3sinx + 1 = 0
(2sinx - 1)(sinx - 1) = 0
sin x = ½ or sin x = 1
x = p/6, 5p/6, p/2

Remember, if sinx = 1 then x = p/2 or 5p/2 or 9p/2 or ... and similarly for arcsin(½). In this question, you are asked for values of x between 0 and p. You must write down all of the appropriate solutions.