5.9 The forms r cos(theta +/- alpha) and r sin(theta +/- alpha) — Learn

Prerequisites: the addition formulae; exact values; solving a simple trigonometric equation; Pythagoras' theorem.

By the end of this section you can: write $a\cos\theta + b\sin\theta$ as a single cosine or sine; find its greatest and least values; use the form to solve an equation.

1. One wave from two

A sum of a cosine and a sine of the same angle is itself a single wave. For constants $a$ and $b$,

where

The value of $R$ comes from expanding the right-hand side with the addition formula:

Matching this with $a\cos\theta + b\sin\theta$ gives

Squaring and adding these two equations gives $a^2 + b^2 = R^2$, so $R = \sqrt{a^2 + b^2}$. Dividing the second by the first gives $\tan\alpha = \dfrac{b}{a}$.

The angle $\alpha$ must lie in the quadrant where $\cos\alpha$ has the sign of $a$ and $\sin\alpha$ has the sign of $b$. The single value returned by $\arctan$ is not always the right one, so check the signs of $a$ and $b$.

You can also write the same sum as a shifted sine:

where $R = \sqrt{a^2 + b^2}$ and $\sin\beta = \dfrac{a}{R}$, $\cos\beta = \dfrac{b}{R}$.

2. The greatest and least values

Because a cosine lies between $-1$ and $1$,

So the greatest value of $a\cos\theta + b\sin\theta$ is $R$, and the least value is $-R$. This is useful in modelling: the single number $R$ gives the peak of the quantity without drawing a graph.

Example 1. Write $3\cos\theta + 4\sin\theta$ in the form $R\cos(\theta - \alpha)$, where $\alpha$ is an acute angle.

Both $3$ and $4$ are positive, so $\alpha$ is acute and the angle from the calculator is correct. Therefore

The greatest value is $5$ and the least is $-5$.

Example 2. Write $-\cos\theta + \sqrt{3}\sin\theta$ in the form $R\cos(\theta - \alpha)$.

Here $a = -1$ and $b = \sqrt{3}$. The angle $\alpha$ must have a negative cosine and a positive sine, so it lies in the second quadrant. The related acute angle is $\arctan(\sqrt{3}) = \dfrac{\pi}{3}$, so

Therefore

3. Solving an equation with the single-wave form

The single-wave form turns $a\cos\theta + b\sin\theta = c$ into one cosine equation, which is much easier to solve.

Example 3. Solve $3\cos\theta + 4\sin\theta = 2$ for $0 \le \theta < 2\pi$.

From Example 1, the left-hand side is $5\cos(\theta - 0.9273)$. So

The principal value is $\cos^{-1}(0.4) \approx 1.1593$ radians. Cosine is also positive at the negative of this angle, so

Only the first lies in $0 \le \theta < 2\pi$. The second is negative, so add $2\pi$:

The solutions are $\theta \approx 2.087$ and $\theta \approx 6.051$.

4. Try it: the single-wave explorer

Predict first. You will change $a$ and $b$ in $a\cos\theta + b\sin\theta$. Before you touch the controls, predict what the greatest value becomes when $a = 3$ and $b = 4$, and whether changing the sign of $b$ moves the peak left or right. Write both predictions down.

Test. Move the sliders and watch the sum wave. Compare the peak height with the value of $R$ in the readout. Set $a = 0$ and check that the result is a pure sine wave.

Explain. Write two sentences. The first explains why the peak height depends on both $a$ and $b$ through $R = \sqrt{a^2 + b^2}$. The second explains why the peak slides along the $\theta$-axis when $b$ changes.

Watch out. Two mistakes are common with the single-wave form.

Using the wrong quadrant for $\alpha$. The calculator gives one angle for $\tan\alpha = \dfrac{b}{a}$. If $a$ is negative, the true $\alpha$ is in the second or third quadrant, not the one a calculator returns by default. Check the signs of $a$ and $b$ and adjust $\alpha$ by $\pi$ when needed.

Mixing up the forms. In $R\cos(\theta - \alpha)$, the matching gives $a = R\cos\alpha$ and $b = R\sin\alpha$. In $R\sin(\theta + \beta)$ the roles swap. Decide which form the question asks for before you match coefficients.

Re-check: expand your answer with the addition formula and confirm that the coefficients of $\cos\theta$ and $\sin\theta$ match the original $a$ and $b$.

Progress check

Return to the chapter problem. The tide height is $h = 12\cos\theta + 5\sin\theta$ metres.

1. Write $h$ in the form $R\cos(\theta - \alpha)$.
2. State the greatest height the water reaches.
3. Explain why the greatest height is not $12 + 5 = 17$ m.

You should now be able to

• write $a\cos\theta + b\sin\theta$ as $R\cos(\theta - \alpha)$ or $R\sin(\theta + \beta)$;
• find $R$ and $\alpha$, choosing the correct quadrant for $\alpha$;
• state the greatest and least values of the sum;
• solve an equation by reducing it to a single cosine or sine;
• check the form by expanding it again.