7.4 General solutions to trigonometric equations
When is this equation true?
Using the spangles (in the previous section), we know that it’s true when . In the interval in question, however, this isn’t the only point where it’s true! If you look at the graph of , it’s clear that there’s another solution to this in the interval . Because of ’s symmetry, we know that this solution will be at .
We can generalise this to any point (and for different trig functions). For any point which is in the range of , we know
We can call the value returned by any inverse trig function the \sayprinciple value. It is usually the angle closest to . However, this value is not necessarily the only possible value. For one, we know that and repeat every (or ) so for any value of which solves or , then that value plus or minus any multiple of (or ) will also solve the equation.
The other thing to bear in mind is that has an axis of symmetry in the lines and . This means that if is the principal value solving , then is also a solution.
For , something very similar is the case, except that the axis of symmetry is in the line , and thus if is the principal value solving , then is also a solution.
Overall, we can write that for
And that for
This is also equivalent to