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 take from each side to get .

This factorises to give

Either or

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 substituting to get and 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

.

Subtract from both sides to get

.

is a common factor so we can factorise to get .

Either or .

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

. Divide both sides by :