Minimum and Maximum Values

Video lesson · from 1:01:141 worked example

One lesson video covers all of Trigonometry, so it opens at 1:01:14 for this topic — not from the beginning.

Theory

By expressing $f(x) = p\cos x + q\sin x$ as a single trigonometric function, we can find the minimum and maximum values of the function and the corresponding angles at which these values occur.

It is useful to remember the minimum and maximum values of $f(x) = \sin x$ and $f(x) = \cos x$ .

$\sin x$ reaches its maximum of 1 at $90^\circ (\pi/2)$ and its minimum of -1 at $270^\circ (3\pi/2)$ .
$\cos x$ reaches its maximum of 1 at $0^\circ$ or $360^\circ (2\pi)$ and its minimum of -1 at $180^\circ (\pi)$ .

⚠️ Common Examiner Traps

Read them straight off: once in the form $k\cos(x-\alpha)$ , the maximum is $k$ and the minimum is $-k$ . No calculus is needed, and using calculus here wastes time.
Give the value and where it happens if asked: the maximum occurs when the bracket is zero, so $x = \alpha$ . Read whether the question wants the value, the angle, or both.
A vertical shift moves both: for $k\cos(x-\alpha) + d$ the maximum is $k + d$ and the minimum is $-k + d$ .
Check the answer sits in the domain: if the stated range does not include $x = \alpha$ , the maximum will be at an end of the interval instead.

Worked examples

Example 1

Write $\cos x + 4\sin x$ in the form $k\cos(x - \alpha)$ where $k > 0$ and $0 \leq \alpha \leq 2\pi$ , and state the minimum and maximum values and the values of $x$ at which they occur.

1. Rewrite

k\cos(x - \alpha) = k\cos x\cos\alpha + k\sin x\sin\alpha

\begin{aligned} k\cos\alpha &= 1 \\ k\sin\alpha &= 4 \end{aligned}

k = \sqrt{1^2 + 4^2} = \sqrt{17}

\tan\alpha = \frac{4}{1} = 4

Base angle $\alpha \approx 1.326$ rad (Q1). Then the function is $\sqrt{17}\cos(x - 1.326)$ .

2. Maximum and minimum:

Max value is $\sqrt{17}$ , Min value is $-\sqrt{17}$ .

Max occurs when $\cos(x - 1.326) = 1$ . From graph, $\cos 0 = 1$ .

\begin{aligned} x - 1.326 &= 0 \\ x &= 1.326 \end{aligned}

Min occurs when $\cos(x - 1.326) = -1$ . From graph, $\cos\pi = -1$ .

\begin{aligned} x - 1.326 &= \pi \\ x &= \pi + 1.326 \\ x &\approx 4.468 \end{aligned}