9.1 Q-10

Exercise 9.1 — Question 10

This question involves finding the maximum and minimum values of trigonometric functions.

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Key Concepts

Maximum and Minimum of $y=a+b\sin (cx+d)$ or $y=a+b\cos (cx+d)$

Since $\sin$ and $\cos$ always satisfy $-1\le \sin (\cdot )\le 1$ and $-1\le \cos (\cdot )\le 1$:

$y_{\max }=a+|b|,y_{\min }=a-|b|$

Example: For $y=5-3\cos (2\theta )$:

• Here $a=5$, $b=-3$, so $|b|=3$
• $y_{\max }=5+3=8$ (when $\cos (2\theta )=-1$)
• $y_{\min }=5-3=2$ (when $\cos (2\theta )=1$)

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Maximum and Minimum of $y=\frac{1}{f(x)}$ when $f(x)>0$

When $f(x)>0$ for all $x$, the reciprocal relationship reverses the inequality:

$y_{\max }=\frac{1}{\min f(x)},y_{\min }=\frac{1}{\max f(x)}$

This is because a smaller denominator produces a larger fraction, and vice versa.

Example: For $y=\frac{1}{2+\sin \theta }$:

• $f(\theta )=2+\sin \theta$, with $\sin \theta \in [-1,1]$
• $\min f(\theta )=2+(-1)=1$, so $y_{\max }=\frac{1}{1}=1$
• $\max f(\theta )=2+1=3$, so $y_{\min }=\frac{1}{3}$
• Range: $\left[\frac{1}{3},1\right]$

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Domain and Range Reminder

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