Solving Basic Equations

It is important to write down all of the solutions in the interval that you are given (see the page on sin, cos and tan).

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.