Ranges and Extrema of R-form | CAIE 9709 Pure Mathematics 2 Notes

Pure Mathematics 2

Mathematics9709

Find both the extreme value and where it occurs

With $R>0$ , a full wave $R\sin(\theta+\alpha)$ reaches $R$ when its sine is 1, and $-R$ when its sine is −1. An added constant shifts both extremes by that amount.

Worked example

Find the maximum and minimum of 2 + 3 sin θ + 4 cos θ for 0° ≤ θ ≤ 360°, and the angles giving them.

Write it as $2+5\sin(\theta+\alpha)$ , where $\alpha=\tan^{-1}(4/3)$ .

\begin{gathered}\max=7\\\text{at }\theta=90^\circ-\alpha\\\min=-3\\\text{at }\theta=270^\circ-\alpha\end{gathered}

The angles are approximately $36.9^\circ$ and $216.9^\circ$ . Both lie in the interval, so both extreme values are attained.

Your turnPractice

Find the range of $1+\sqrt3\cos x+\sin x$ for $0^\circ\le x\le360^\circ$ , and an angle at each extreme.

Show worked answer

Answer

-1\le y\le3

Write $y=1+2\cos(x-30^\circ)$ . The maximum is at $x=30^\circ$ and the minimum at $x=210^\circ$ .

A restricted interval may not contain the peak

Do not automatically report $-R\le y\le R$ on a short interval. On a closed interval, where both endpoints are included, find where the shifted sine or cosine reaches 1 or −1, keep only those angles inside the interval, and compare their outputs with both endpoints. Between consecutive peaks and troughs the wave moves in one direction, so these are the candidates for extremes. If an endpoint is excluded, its output can be a boundary without belonging to the range.

Move the right endpoint. Notice when the peak first enters the interval and when the trough enters.

Right endpoint L = 90°

Minimum ≈ 3 at θ ≈ 90°.

Maximum ≈ 5 at θ ≈ 36.87°.

Blue: selected interval. Grey dashed: outside it. Green: maximum. Purple: minimum.

Worked example

Find the range of 3 sin θ + 4 cos θ for 0° ≤ θ ≤ 30°.

The full-wave peak is at approximately $36.9^\circ$ , outside this interval. The wave is still increasing throughout it. Compare the endpoints.

\begin{gathered}f(0^\circ)=4\\f(30^\circ)=\frac32+2\sqrt3\end{gathered}

Answer

4\le f(\theta)\le\frac32+2\sqrt3

Your turnPractice

Find the exact range of $3\sin\theta+4\cos\theta$ for $0^\circ\le\theta\le90^\circ$ .

Show worked answer

Answer

3\le y\le5

The peak at $\theta=90^\circ-\alpha$ is included. The endpoint values are 4 and 3; the trough is not included.

Find both the extreme value and where it occurs

With $R>0$ , a full wave $R\sin(\theta+\alpha)$ reaches $R$ when its sine is 1, and $-R$ when its sine is −1. An added constant shifts both extremes by that amount.

Worked example

Find the maximum and minimum of 2 + 3 sin θ + 4 cos θ for 0° ≤ θ ≤ 360°, and the angles giving them.

Write it as $2+5\sin(\theta+\alpha)$ , where $\alpha=\tan^{-1}(4/3)$ .

\begin{gathered}\max=7\\\text{at }\theta=90^\circ-\alpha\\\min=-3\\\text{at }\theta=270^\circ-\alpha\end{gathered}

The angles are approximately $36.9^\circ$ and $216.9^\circ$ . Both lie in the interval, so both extreme values are attained.

Your turnPractice

Find the range of $1+\sqrt3\cos x+\sin x$ for $0^\circ\le x\le360^\circ$ , and an angle at each extreme.

Show worked answer

Answer

-1\le y\le3

Write $y=1+2\cos(x-30^\circ)$ . The maximum is at $x=30^\circ$ and the minimum at $x=210^\circ$ .

A restricted interval may not contain the peak

Move the right endpoint. Notice when the peak first enters the interval and when the trough enters.

Right endpoint L = 90°

Minimum ≈ 3 at θ ≈ 90°.

Maximum ≈ 5 at θ ≈ 36.87°.

Blue: selected interval. Grey dashed: outside it. Green: maximum. Purple: minimum.

Worked example

Find the range of 3 sin θ + 4 cos θ for 0° ≤ θ ≤ 30°.

The full-wave peak is at approximately $36.9^\circ$ , outside this interval. The wave is still increasing throughout it. Compare the endpoints.

\begin{gathered}f(0^\circ)=4\\f(30^\circ)=\frac32+2\sqrt3\end{gathered}

Answer

4\le f(\theta)\le\frac32+2\sqrt3

Your turnPractice

Find the exact range of $3\sin\theta+4\cos\theta$ for $0^\circ\le\theta\le90^\circ$ .

Show worked answer

Answer

3\le y\le5

The peak at $\theta=90^\circ-\alpha$ is included. The endpoint values are 4 and 3; the trough is not included.