## Content

### Formulas

A formula is a way of relating different quantities using algebra. The well-known formula

$A = \pi r^2$

relates the area $$A$$ of a circle to its radius $$r$$. The variable $$A$$ is the subject of the formula since it stands alone on the left-hand side of the equation.

It is often the case in mathematics that we need to rearrange a formula and make one of the other variables to be the subject.

For example, the cosine rule states that if $$a,b,c$$ are the side lengths of a triangle and $$A$$ is the angle opposite $$a$$, then

$a^2=b^2+c^2-2bc\,\cos A.$

If we wish to find the angle in a triangle, given the three side lengths, then we need to make $$\cos A$$ the subject. Rearranging the terms, we have

$2bc\,\cos A = b^2+c^2-a^2 \implies \cos A = \dfrac{b^2+c^2-a^2}{2bc}.$

Students should be well-practised in this sort of procedure.

##### Exercise 10

The compound interest formula states that $$A = P(1 + R)^n$$, where $$P$$ is the principal, $$R$$ the interest rate per unit time, $$n$$ the number of units of time, and $$A$$ the amount to which the principal has increased. Make $$n$$ the subject of the formula.

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