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
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 :