Question

How can you do these types of problems quickly?

For A for example I know that one answer is pi/6 but how would I quickly know that the other solution is 5pi/6 without having to draw the unit circle?
Is there a shortcut to find the two solutions quickly when it asks for the solutions from 0 to 2pi? It seems very impractical to have to write down the entire unit circle just to solve these types of problems.
One other way is to use a calculator but my quiz doesn't allow me to use a calculator for these types of problems

Answer 1

Please see below.

Explanation:

There are three things required for this.

1. You should know the trigonometric ratios, at least of angles relating to first quadrant and if possible remember well the entire unit circle of trigonometric ratios (shown below).
   
2. You should also know in which quadrants each of the trigonometric ratios are positive (in others, it will be negative). Well every ratio is positive in `Q1`, but sine and cosecant ratios are positive in `Q2` too, tangent and cotangent are positive in `Q3` too and cosine and secant are positive in `Q4` too.

For `Q2` subtract angle from `180^@`, for `Q3` add angle to `180^@` and for `Q4` subtract angle from `360^@` or `0^@`.

For example, for `sinx=1/2`, we know `x=30^@`, but as sine ratio is positive in `Q2` too, other angle would be `180^@-30^@=150^@` too. Similarly `2tanx-5=1` means `tanx=3` and so if `tanalpha=3` and alpha is in `Q1`, the angle in `Q3` would be `alpha+180^@`. Similarly as `cosx=sqrt2/2=cos45^@` and hence `x=45^@` and `360^@-45^@=315^@`, as cosine ratio is positive in `Q4`.

The above should serve the purpose most of the time. However, last but not the least, if possible, remember the pattern (or graph) of the trigonometric ratios at least in the range `[0,2i]`.