Papers 2 and 3. Any combination \(a\sin x + b\cos x\) is a single sine wave, shifted and stretched. Writing it as \(R\sin(x + \alpha)\) or \(R\cos(x - \alpha)\) lets you solve equations in one step and read off greatest and least values at a glance.
\(a\sin x + b\cos x \equiv R\sin(x + \alpha)\) where \(R = \sqrt{a^2 + b^2}\), \(R\cos\alpha = a\), \(R\sin\alpha = b\), so \(\tan\alpha = \dfrac{b}{a}\). Expand the target form, match coefficients of \(\sin x\) and \(\cos x\), then square-and-add for \(R\) and divide for \(\alpha\).
Fig. 1: y = 3 sin x + 4 cos x is the single wave y = 5 sin(x + 53.1°): amplitude 5, maximum at x = 90° − 53.1° = 36.9°. It meets y = 2 at x = 103.3° and 330.4°
Section 01
Matching coefficients.
Fig. 1 shows that \(3\sin x + 4\cos x\) is just a sine wave of amplitude 5, shifted left. To find the amplitude \(R\) and the shift \(\alpha\), expand the form you want and match it term by term with the expression you have.
Expression
Usual target
Match
\(a\sin x + b\cos x\)
\(R\sin(x + \alpha)\)
\(R\cos\alpha = a,\ R\sin\alpha = b\)
\(a\sin x - b\cos x\)
\(R\sin(x - \alpha)\)
\(R\cos\alpha = a,\ R\sin\alpha = b\)
\(a\cos x + b\sin x\)
\(R\cos(x - \alpha)\)
\(R\cos\alpha = a,\ R\sin\alpha = b\)
\(a\cos x - b\sin x\)
\(R\cos(x + \alpha)\)
\(R\cos\alpha = a,\ R\sin\alpha = b\)
With \(a, b > 0\) these choices make \(\alpha\) acute, which is what questions usually ask for.
R: square and add, \(R^2(\cos^2\alpha + \sin^2\alpha) = a^2 + b^2\), so \(R = \sqrt{a^2 + b^2}\). Leave it as a surd if it is not a whole number.
α: divide, \(\tan\alpha = \dfrac{R\sin\alpha}{R\cos\alpha}\). Give degrees to 2 d.p. if you will use it again.
Worked example R sin(x + α)
Express \(3\sin x + 4\cos x\) in the form \(R\sin(x + \alpha)\), where \(R > 0\) and \(0^\circ < \alpha < 90^\circ\).
\(\tan\alpha = \dfrac{R\sin\alpha}{R\cos\alpha}\), so it is the coefficient matched with \(\sin\alpha\) on top. For \(3\sin x + 4\cos x = R\sin(x + \alpha)\), \(\tan\alpha = \dfrac{4}{3}\), not \(\dfrac{3}{4}\). Always write the two matching equations first.
Section 02
Solving equations.
Once the left side is \(R\sin(x + \alpha)\), the equation has only one trig term. The one new skill is adjusting the range: if \(0^\circ \le x \le 360^\circ\), then \(x + \alpha\) runs from \(\alpha\) to \(360^\circ + \alpha\).
Worked example Degrees
Solve \(3\sin x + 4\cos x = 2\) for \(0^\circ \le x \le 360^\circ\).
Range: \(53.13^\circ \le x + 53.13^\circ \le 413.13^\circ\).
\(\sin^{-1} 0.4 = 23.58^\circ\), which is too small. The others: \(180^\circ - 23.58^\circ = 156.42^\circ\) and \(360^\circ + 23.58^\circ = 383.58^\circ\).
Subtract \(53.13^\circ\): \(x = 103.29^\circ\) and \(x = 330.45^\circ\).
Range: \(-\dfrac{\pi}{3} < \theta - \dfrac{\pi}{3} < \dfrac{5\pi}{3}\). Here \(\theta - \dfrac{\pi}{3} = \dfrac{\pi}{4}\) or \(-\dfrac{\pi}{4}\) (\(\dfrac{7\pi}{4}\) is too big).
Solve for the whole bracket \(x + \alpha\) in its shifted range FIRST, and only then subtract \(\alpha\). Taking the calculator value \(23.58^\circ\) and subtracting \(53.13^\circ\) gives a negative angle, and students often then miss \(330.4^\circ\).
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
Quick check+10 XP
\(5\sin x + 12\cos x \equiv R\sin(x + \alpha)\) with \(R > 0\). What is \(R\)?
Quick check+10 XP
For \(a\sin x + b\cos x \equiv R\sin(x + \alpha)\), which is true?
Quick check+10 XP
What is the greatest value of \(6\sin x + 8\cos x\)?
Quick check+10 XP
Which form is most natural for \(4\cos x + 3\sin x\) with an acute \(\alpha\)?
Section 03
Greatest and least values.
Because \(-1 \le \sin(x + \alpha) \le 1\), the expression \(R\sin(x + \alpha)\) lies between \(-R\) and \(R\). The greatest value happens when \(x + \alpha = 90^\circ\), the least when \(x + \alpha = 270^\circ\).
Worked example Greatest and least of a fraction
Find the greatest and least values of \(\dfrac{20}{7 + 3\sin x + 4\cos x}\), and the values of \(x\) in \(0^\circ \le x \le 360^\circ\) where they occur.
The denominator is \(7 + 5\sin(x + 53.13^\circ)\), which ranges from \(7 - 5 = 2\) to \(7 + 5 = 12\).
The fraction is greatest when the denominator is least: \(\dfrac{20}{2} = 10\), when \(\sin(x + 53.13^\circ) = -1\), so \(x + 53.13^\circ = 270^\circ\), \(x = 216.9^\circ\).
The fraction is least when the denominator is greatest: \(\dfrac{20}{12} = \dfrac{5}{3}\), when \(x + 53.13^\circ = 90^\circ\), \(x = 36.9^\circ\).
Answer: Greatest \(10\) at \(x = 216.9^\circ\); least \(\dfrac{5}{3}\) at \(x = 36.9^\circ\)
Examiner trap
For a fraction with the trig expression underneath, the greatest value of the fraction comes from the LEAST value of the denominator. Also check the denominator can never be zero.
Stage 2 · Exam
Exam-style questions
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Exam style+20 XP
Express \(2\sin x + 3\cos x\) in the form \(R\sin(x + \alpha)\), with \(R > 0\) and \(0^\circ < \alpha < 90^\circ\).
Exam style+20 XP
Find the greatest value of \(6\cos x + 8\sin x + 3\) and the value of \(x\), \(0^\circ \le x \le 360^\circ\), at which it occurs.
Exam style+20 XP
Solve \(\sqrt{3}\sin x + \cos x = 1\) for \(0^\circ \le x \le 360^\circ\).
Exam style+20 XP
Find the greatest value of \(\dfrac{30}{8 + 3\sin x - 4\cos x}\).
Skill unlocked
One wave, one equation.
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