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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