How can you do these types of problems quickly?

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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

1 Answer
Oct 30, 2017

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).
    https://www.youtube.com/watch?v=6Qv_bPlQS8Ehttps://www.youtube.com/watch?v=6Qv_bPlQS8E
  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].