Chapter 4.3, Problem 93E

(a)

To determine

To fill: The value of each trigonometry function in the table.

Answer to Problem 93E

The complete table shown below.

Explanation of Solution

Given Information:

The given table is as follows.

$\theta$	$0°$	$20°$	$40°$	$60°$	$80°$
$\sin \theta$
$\cos \theta$
$\tan \theta$

Calculation:

Complete the table by using graphing utility.

$\theta$	$0°$	$20°$	$40°$	$60°$	$80°$
$\sin \theta$	0	0.3420	0.6428	0.8660	0.9848
$\cos \theta$	1	0.9397	0.7660	0.5	0.1736
$\tan \theta$	0	0.3640	0.8391	1.7321	5.6713

Therefore, the complete table shown above.

(b)

To determine

The classification of the three trigonometric functions as increasing or decreasing table values.

Answer to Problem 93E

The value of $\sin \theta$ , $\cos \theta$ and $\tan \theta$ are classified as increasing, decreasing and increasing.

Explanation of Solution

Given Information:

Consider the following table.

$\theta$	$0°$	$20°$	$40°$	$60°$	$80°$
$\sin \theta$	0	0.3420	0.6428	0.8660	0.9848
$\cos \theta$	1	0.9397	0.7660	0.5	0.1736
$\tan \theta$	0	0.3640	0.8391	1.7321	5.6713

Calculation:

As shown in the table, the value of $\sin \theta$ is increasing. So, it can be classified as increasing function.

As shown in the table, the value of $\cos \theta$ is decreasing. So, it can be classified as decreasing function.

As shown in the table, the value of $\tan \theta$ is increasing. So, it can be classified as increasing function.

Therefore, value of $\sin \theta$ , $\cos \theta$ and $\tan \theta$ are classified as increasing, decreasing and increasing.

(c)

To determine

To verify: The tangents function is the quotient of the sin and cosine function.

Explanation of Solution

Given Information:

Consider the following table.

$\theta$	$0°$	$20°$	$40°$	$60°$	$80°$
$\sin \theta$	0	0.3420	0.6428	0.8660	0.9848
$\cos \theta$	1	0.9397	0.7660	0.5	0.1736
$\tan \theta$	0	0.3640	0.8391	1.7321	5.6713

Formula Used:

From the trigonometry ratios,

  $\tan \theta =\frac{\text{sin}\theta }{\cos \theta }$

Proof:

Consider the angle $0°$ for verification.From the table, the value of $\tan 0°$ is 0.

Calculate the ratio of $\frac{\text{sin}\theta }{\cos \theta }$ .

  $\begin{array}{c}\tan 0°=\frac{\text{sin}0°}{\cos 0°}\\ 0=\frac{0}{1}\\ 0=0\end{array}$

From the above, it is clear that the left hand side is equal to the right hand side.

Hence, proved.