We have seen that you can find trig ratios for any angle, by determining which acute angle it is equivalent to and then deciding whether the function should be positive or negative, depending on the quadrant. The results are summarized below.

The implication of this is that there will be more than one solution to the equation

In degrees, we know that x could equal 30^o, but there is also a positive value of sin in the second quadrant, where the rule is that you subtract the acute angle from 180. Therefore 150 is also a solution of our equation.

There are in fact an infinite number of solutions to the equation

Therefore, when ever you are asked to solve a trigonometric equation, you will need to find all the values in a given range.

For example, find all the solutions of the equation in the given range.

First find the first solution ( in the first quadrant).

To find the second result, refer to the table above (which you should memorize). You can see that cos also has a positive value in the third quadrant, which you find by adding 180^o

Therefore the second solution is

so it is unlikely that we have found all of our solutions.

In general it is a good idea to continue finding solutions until you find a solution outside the range. This is easy to do, because the trig curves are periodic. This means that they keep repeating themselves exactly. If you look at the graph of cos x, you will see that it's period is

This gives the following list of solutions

The last solution is outside the required range, so the answer to our question is

Exactly the same method can be used to find all the required solutions for an equation using sin and tan, with the exception that tan has a period of 180^o.

On the next page there will be some more examples, covering the solutions of equations such as

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