The R-formula (a cos + b sin)AQA A-Level Maths: Revision notes

An expression $a\cos\theta+b\sin\theta$ can always be written as a single trigonometric function. To write it as $R\cos(\theta-\alpha)$, expand with the compound angle formula: $R\cos(\theta-\alpha)=R\cos\alpha\cos\theta+R\sin\alpha\sin\theta.$ Comparing coefficients, $R\cos\alpha=a$ and $R\sin\alpha=b$, so $R=\sqrt{a^2+b^2},\qquad\tan\alpha=\frac{b}{a}.$ For $3\cos\theta+4\sin\theta$: $R=5$, $\tan\alpha=\frac43$, $\alpha=53.1^\circ$, so $3\cos\theta+4\sin\theta=5\cos(\theta-53.1^\circ)$.

You can choose any of four equivalent forms. For $a\cos\theta+b\sin\theta$, expand each and match coefficients:

• $R\cos(\theta-\alpha)$: $R\cos\alpha=a$, $R\sin\alpha=b$.
• $R\cos(\theta+\alpha)$: $R\cos\alpha=a$, $R\sin\alpha=-b$.
• $R\sin(\theta+\alpha)$: $R\sin\alpha=a$, $R\cos\alpha=b$.
• $R\sin(\theta-\alpha)$: $R\sin\alpha=-a$, $R\cos\alpha=b$. Example: $\sqrt3\sin\theta-\cos\theta=R\sin(\theta-\alpha)$ gives $R\cos\alpha=\sqrt3$, $R\sin\alpha=1$, so $R=2$, $\alpha=30^\circ$: $2\sin(\theta-30^\circ)$. For $5\sin x+12\cos x=13\sin(x+67.4^\circ)$, $\alpha$ is the angle with $\tan\alpha=\frac{12}{5}$.

Since $-1\le\cos(\theta-\alpha)\le1$, the expression $a\cos\theta+b\sin\theta$ has maximum $R$ and minimum $-R$. The maximum for $3\cos\theta+4\sin\theta$ is $5$ at $\theta=\alpha=53.1^\circ$, and the minimum is $-5$ at $\theta=233.1^\circ$. For $10+3\cos\theta+4\sin\theta$ the greatest value is $10+5=15$ and the least is $10-5=5$. In a model such as a tide depth, $R$ is the amplitude of the variation.

To solve $a\cos\theta+b\sin\theta=c$, write the left side as $R\cos(\theta-\alpha)$, so $\cos(\theta-\alpha)=\frac{c}{R}$. A solution exists only if $|c|\le R$. Work out the range of $\theta-\alpha$ for the interval before listing solutions. Example: $3\cos\theta+4\sin\theta=2.5$ for $0^\circ\le\theta<360^\circ$: $5\cos(\theta-53.1^\circ)=2.5$, so $\cos(\theta-53.1^\circ)=0.5$ and $\theta-53.1^\circ=\pm60^\circ$, giving $\theta=113.1^\circ$ or $-6.9^\circ+360^\circ=353.1^\circ$. Example: $5\sin x+12\cos x=6.5$: $13\sin(x+67.4^\circ)=6.5$, so $x+67.4^\circ=150^\circ$ or $390^\circ$ and $x=82.6^\circ$ or $322.6^\circ$.

Periodic models, such as tides, often take the form $D=k+a\cos(\omega t)+b\sin(\omega t)$. Rewriting as $D=k+R\cos(\omega t-\alpha)$ shows that the depth oscillates between $k-R$ and $k+R$, with the first maximum when $\omega t=\alpha$. Example: $D=10+3\cos(30t)^\circ+4\sin(30t)^\circ=10+5\cos(30t-53.1)^\circ$. Maximum depth $15$ m at $t=1.77$ h. $D=12$ when $\cos(30t-53.1)^\circ=0.4$, that is $t=3.99$ and $t=11.56$, so the depth is below $12$ m for about $7.57$ hours.