You may be asked to solve trig equations, that first need to be re-arranged using the identities that we have looked at. For example, how do you find all of the solutions to the equation

If we are going to solve the equation, then it can contain sin or cos, but not both. If we use the Pythagoras' relation

we can replace the sin with cos

You now just solve the two equations, for the given range of x.

You could find the first solution by using a calculator, but it is not necessary. Since we are dealing with cos and the answer is negative, we know that the first solution will be in the second quadrant. To find the angle we can first find the acute angle that we can work with. If the cos was positive, then the answer would be 60^o. The corresponding angle in the second quadrant is found by subtracting this from 180^o. The first solution is therefore

The second negative cos solution lies in the third quadrant, and is found by adding our acute angle to 180^o. The second solution is therefore

Now solve the second equation

This is where the range is very important. The range that has been given includes both 0 and 360. Since cos repeats every 360^o there will also be a solution at 360^o. We can now combine the results.

The solutions to the equation

are

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