By the end you can
Write $a\sin x + b\cos x$ as $R\sin(x + \alpha)$ or a related form.
Read off the amplitude, maximum and minimum.
Solve $a\sin x + b\cos x = c$ and related inequations.
One wave from two
Add any multiple of $\sin x$ to any multiple of $\cos x$ and the result is still a single sine wave, just stretched vertically and shifted sideways.
y = a sin x + b cos x Turn on JavaScript to use this interactive graph.
Orange: the sum. Blue dashed: y = a sin x. Pink dashed: y = b cos x. The orange curve is always a sine wave with amplitude R, shifted left by α.
Try this
Check with $a = 3$, $b = 4$ that the highest point of the orange curve is at height $5$.
The auxiliary angle method
For $a > 0$:
$$a\sin x + b\cos x = R\sin(x + \alpha), \quad R = \sqrt{a^2 + b^2}, \quad \cos\alpha = \frac{a}{R}, \; \sin\alpha = \frac{b}{R}.$$
The same method gives the forms $R\sin(x - \alpha)$, $R\cos(x + \alpha)$ and $R\cos(x - \alpha)$: expand the target form and match coefficients.
Why it works
$R\sin(x + \alpha) = R\cos\alpha\sin x + R\sin\alpha\cos x$. Matching with $a\sin x + b\cos x$ needs $R\cos\alpha = a$ and $R\sin\alpha = b$. Square and add: $R^2 = a^2 + b^2$. Divide: $\tan\alpha = \dfrac{b}{a}$.
Write $\sqrt3\sin x + \cos x$ in the form $R\sin(x + \alpha)$, and state its maximum and minimum.
$\alpha = 30^\circ$, so $\sqrt3\sin x + \cos x = 2\sin(x + 30^\circ)$.
cos α = √3/2 and sin α = 1/2.
Maximum $2$, minimum $-2$.
Solving equations
Solve $\sqrt3\sin x + \cos x = 1$ for $0^\circ \le x \le 360^\circ$.
$2\sin(x + 30^\circ) = 1$, so $\sin(x + 30^\circ) = \tfrac12$.
$x + 30^\circ = 30^\circ, 150^\circ, 390^\circ$
Shift the domain too: 30° ≤ x + 30° ≤ 390°.
$x = 0^\circ, 120^\circ, 360^\circ$
Common mistake
Forgetting to shift the domain. If $0^\circ \le x \le 360^\circ$, then $x + 30^\circ$ runs from $30^\circ$ to $390^\circ$, and missing $390^\circ$ loses the solution $x = 360^\circ$.
Example 3: an approximate auxiliary angle ●●● Solve $3\sin x + 4\cos x = 2$ for $0^\circ \le x \le 360^\circ$, to the nearest $0.1^\circ$.
$R = 5$ and $\tan\alpha = \dfrac43$, so $\alpha \approx 53.13^\circ$.
$\sin(x + \alpha) = 0.4$, with $53.13^\circ \le x + \alpha \le 413.13^\circ$.
$x + \alpha \approx 156.42^\circ$ or $383.58^\circ$
sin⁻¹ 0.4 ≈ 23.58°. In range: 180° − 23.58° and 360° + 23.58°.
$x \approx 103.3^\circ$ or $x \approx 330.4^\circ$
Inequations
Solve $\sqrt3\sin x + \cos x > 1$ for $0^\circ \le x \le 360^\circ$.
$\sin(x + 30^\circ) > \tfrac12$ with $30^\circ \le x + 30^\circ \le 390^\circ$.
$30^\circ < x + 30^\circ < 150^\circ$
Sine is above ½ strictly between 30° and 150° in this range.
$0^\circ < x < 120^\circ$
$R$ for $asin x + bcos x$
$\sqrt{a^2 + b^2}$
$lpha$ in $Rsin(x + lpha)$
$\cos\alpha = \dfrac{a}{R}$, $\sin\alpha = \dfrac{b}{R}$, so $\tan\alpha = \dfrac{b}{a}$.
Maximum of $asin x + bcos x$
$\sqrt{a^2 + b^2}$
$sqrt3sin x + cos x$
$2\sin(x + 30^\circ)$
Find $R$ when $3\sin x + 4\cos x = R\sin(x + \alpha)$.
Find the maximum value of $\sqrt3\sin x + \cos x$.
$\sqrt3\sin x + \cos x = $
$2\sin(x + 60^\circ)$ $2\cos(x + 30^\circ)$ $2\sin(x + 30^\circ)$ $4\sin(x + 30^\circ)$ Solve $\sqrt3\sin x + \cos x = 1$ for $0^\circ \le x \le 360^\circ$. Enter the solutions separated by commas.
For $3\sin x + 4\cos x = 5\sin(x + \alpha)$, find $\alpha$ to 1 decimal place of a degree.
Find the minimum value of $\sin x + \cos x$.
$R = \sqrt2$, so the minimum is $-\sqrt2$.
Solve $\sqrt3\sin x + \cos x > 1$ for $0^\circ \le x \le 360^\circ$.
$0^\circ < x < 120^\circ$ $30^\circ < x < 150^\circ$ $120^\circ < x < 360^\circ$ $0^\circ \le x \le 120^\circ$ See Example 4. The endpoints give equality, so they are excluded.
The equation $3\sin x + 4\cos x = 6$:
has two solutions in $0^\circ$ to $360^\circ$ has one solution has infinitely many solutions has no solutions The left side is never more than $5$.
If $0^\circ \le x \le 360^\circ$, then $x + 30^\circ$ lies between:
$0^\circ$ and $360^\circ$ $30^\circ$ and $390^\circ$ $-30^\circ$ and $330^\circ$ $30^\circ$ and $360^\circ$ Add $30^\circ$ to both ends.
$\sin x - \sqrt3\cos x = R\sin(x - \alpha)$. Find $R$.
Try again
New numbers every time. Keep going until you get five in a row.
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