## Solving Basic Equations

This pages shows you how to solve 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 2cos^{2}x + 3sinx = 3, giving your answer in radians for 0< x <p.

2cos^{2}x + 3sinx - 3 = 0

We need to get everything in terms of sinx or everything in terms of cosx. Since we know that cos^{2}x = 1 - sin^{2}x:

2(1 - sin^{2}x) + 3sinx - 3 = 0

2 - 2sin^{2}x + 3sinx - 3 = 0

-2sin^{2}x + 3sinx - 1 = 0

2sin^{2}x - 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.