Chapter 4.3, Problem 84E

a.

To determine

To discuss: the behavior of the sine function for $0°\le \theta \le 90°$ .

Answer to Problem 84E

Sine function increases in the interval $0°\le \theta \le 90°$ .

Explanation of Solution

First use the calculator to complete the given table:

  $\sin 0°=0$

  $\sin 18°\approx 0.31$

  $\sin 36°\approx 0.59$

  $\sin 54°\approx 0.81$

  $\sin 72°\approx 0.95$

  $\sin 90°=1$

Thus,

$\theta$	$0°$	$18°$	$36°$	$54°$	$72°$	$90°$
$\sin \theta$	0	0.31	0.59	0.81	0.95	1

From the above table it is clear that sine function increases in the interval $0°\le \theta \le 90°$ .

b.

To determine

To discuss: the behavior of the cosine function for $0°\le \theta \le 90°$ .

Answer to Problem 84E

Cosine function decreases in the interval $0°\le \theta \le 90°$ .

Explanation of Solution

First use the calculator to complete the given table:

  $\cos 0°=1$

  $\cos 18°\approx 0.95$

  $\cos 36°\approx 0.81$

  $\cos 54°\approx 0.59$

  $\cos 72°\approx 0.31$

  $\cos 90°=0$

Thus,

$\theta$	$0°$	$18°$	$36°$	$54°$	$72°$	$90°$
$\cos \theta$	1	0.95	0.81	0.59	0.31	0

From the above table it is clear that cosine function decreases in the interval $0°\le \theta \le 90°$ .

c.

To determine

To explain: the results of parts (a) and (b).

Answer to Problem 84E

Cosine function decreases in the interval $0°\le \theta \le 90°$ .

Explanation of Solution

Consider,

  

  $\sin \theta =\frac{opp}{hyp}$ and $\cos \theta =\frac{adj}{hyp}$

Now as $\theta$ increases, $\sin \theta$ increases and $\cos \theta$ decreases.

The reason behind it is that the hypotenuse is fixed so on increasing the angle $\theta$ , length of the opposite side increases and the length of the adjacent side decreases.