The R-form: a sin θ + b cos θ in one function

9 minA Level3.3.2

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can write $a\sin\theta + b\cos\theta$ as a single wave like $R\sin(\theta + \alpha)$ , find $R$ and $\alpha$ , read off maximum and minimum values, and solve equations such as $3\cos\theta + 2\sin\theta = 1$ .

Key to the symbols

Symbol	Meaning
$a$ , $b$	the numbers in front of $\sin\theta$ and $\cos\theta$
$R$	the height (amplitude) of the single combined wave; always positive
$\alpha$	"alpha": how far the combined wave is shifted sideways

What they ask

Two kinds of question, both using the same trick:

Find the maximum or minimum of something like $3\sin\theta + 4\cos\theta$ .
Solve an equation like $a\sin\theta + b\cos\theta = c$ . That's awkward with two different functions, but easy once it's one.

The idea: a sine wave plus a cosine wave of the same period is one wave, just taller and shifted sideways.

Where R and α come from

Suppose $a\sin\theta + b\cos\theta = R\sin(\theta + \alpha)$ .

Expand the right side with the addition formula:

$R\sin(\theta + \alpha) = R\cos\alpha\sin\theta + R\sin\alpha\cos\theta$

Match the numbers in front of $\sin\theta$ and $\cos\theta$ :

$a = R\cos\alpha, \qquad b = R\sin\alpha$

To find $R$ : square and add. Since $\cos^2\alpha + \sin^2\alpha = 1$ :

$a^2 + b^2 = R^2 \quad\Rightarrow\quad R = \sqrt{a^2 + b^2}$

To find $\alpha$ : divide the second by the first. The $R$ cancels:

$\tan\alpha = \frac{b}{a}$

Definition

The R-forms

Choose the form whose expansion has the same signs as your expression:

Expression	Write it as	Because the expansion is
$a\sin\theta + b\cos\theta$	$R\sin(\theta + \alpha)$	$R\cos\alpha\sin\theta + R\sin\alpha\cos\theta$
$a\sin\theta - b\cos\theta$	$R\sin(\theta - \alpha)$	$R\cos\alpha\sin\theta - R\sin\alpha\cos\theta$
$a\cos\theta + b\sin\theta$	$R\cos(\theta - \alpha)$	$R\cos\alpha\cos\theta + R\sin\alpha\sin\theta$
$a\cos\theta - b\sin\theta$	$R\cos(\theta + \alpha)$	$R\cos\alpha\cos\theta - R\sin\alpha\sin\theta$

In the cos rows, $a$ is the number in front of $\cos\theta$ and $b$ is the number in front of $\sin\theta$ . In every case (with $a, b > 0$ ): $R = \sqrt{a^2 + b^2}$ and $\tan\alpha = \dfrac{b}{a}$ , with $\alpha$ an acute angle.

Notice the cos forms flip the sign: $\cos(\theta - \alpha)$ has a plus in its expansion. That's why you match the expansion, not the sign inside the bracket.

Maximum and minimum

$\sin(\text{anything})$ is always between $-1$ and $1$ . So $R\sin(\theta + \alpha)$ is always between $-R$ and $R$ .

The maximum is $R$ , when the bracket equals $90°$ .
The minimum is $-R$ , when the bracket equals $270°$ .

For $R\cos(\theta \pm \alpha)$ , the maximum $R$ comes when the bracket is $0°$ (or $360°$ ), and the minimum $-R$ when it is $180°$ .

Adding a constant shifts both. For example, $R\sin(\theta + \alpha) - 7$ has maximum $R - 7$ and minimum $-R - 7$ .

Worked example: Example 1: maximum and minimum

Write $3\sin\theta + 2\cos\theta$ in the form $R\sin(\theta + \alpha)$ , where $R > 0$ and $0° < \alpha < 90°$ . Hence state its maximum and minimum values, and the least value of $3\sin\theta + 2\cos\theta - 7$ .

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Quick check

Multiple choiceQuestion 1 · Apply

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