Trigonometry often reduces to quadratic equations using one of the formulae:

For example, to solve

, we look at the formulae: the second one substitutes for. We obtain.

We takefrom each side to get.

This factorises to give


We may have to use the quadratic formula instead of factorising.

For example,. As before we use the second formula above, obtaining,

This doesn't factorise, but we can find solutions by substitutingto getand solving this with the quadratic formula to find p thence x.

Sometimes the equation we arrive at doesn't look like a quadratic. For example,


For this we use the last of the five formulae above, to get


Subtractfrom both sides to get


is a common factor so we can factorise to get.

Eitheror .

Finally, there may be equations that are not actually quadratic, but still require manipulation.

. Divide both sides by:

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