Chapter 4.3, Problem 87E

a.

To determine

Complete the table.

Answer to Problem 87E

  $\begin{array}{ccccccc}\theta & 0°& 18°& 36°& 54°& 72°& 90°\\ \sin \theta & 0& 0.309& 0.588& 0.809& 0.951& 1\\ \cos \theta & 1& 0.951& 0.809& 0.588& 0.309& 0\end{array}$

Explanation of Solution

Given information:

Complete the table.

  $\begin{array}{ccccccc}\theta & 0°& 18°& 36°& 54°& 72°& 90°\\ \sin \theta & & & & & & \\ \cos \theta & & & & & & \end{array}$

Calculation:

By using calculator following function can be evaluated as follows

  $\begin{array}{l}\sin 0°=0,\sin 18°=0.309,\sin 36°=0.588\\ \cos 0°=1,\cos 18°=0.951,\cos 36°=0.809\end{array}$

Co-functions of complementary angles are equal. If $\theta$ is an acute angle, the following relationships are true

  $\begin{array}{l}\sin (90°-\theta )=\cos \theta \\ \cos (90°-\theta )=\sin \theta \end{array}$

The value of other functions can be found as follows

  $\sin 54°=\cos \left(90°-54°\right)$

  $\sin 54°=\cos 36°$

  $=0.809$

  $\begin{array}{l}\sin 72°=\cos \left(90°-72°\right)\\ \sin 72°=\cos 18°\end{array}$

  $=0.951$

  $\begin{array}{l}\sin 90°=\cos \left(90°-90°\right)\\ \sin 90°=\cos 0°\end{array}$

  $=1$

  $\begin{array}{l}\cos 54°=\sin \left(90°-54°\right)\\ \cos 54°=\sin 36°\end{array}$

  $0.588$

  $\begin{array}{l}\cos 72°=\sin \left(90°-72°\right)\\ \cos 72°=\sin 18°\end{array}$

  $=0.309$

  $\begin{array}{l}\cos 90°=\sin \left(90°-90°\right)\\ \cos 90°=\sin 0°\end{array}$

  $=0$

  $\begin{array}{ccccccc}\theta & 0°& 18°& 36°& 54°& 72°& 90°\\ \sin \theta & 0& 0.309& 0.588& 0.809& 0.951& 1\\ \cos \theta & 1& 0.951& 0.809& 0.588& 0.309& 0\end{array}$

b.

To determine

Discuss the behaviour of the sine function for $\theta$ in the range from $0°$ to $90°$ .

Answer to Problem 87E

The sine function covers its minimum value and maximum value in the given interval.

Explanation of Solution

Given information:

Discuss the behaviour of the sine function for $\theta$ in the range from $0°$ to $90°$ .

Calculation:

From the table, it can be seen that sine function is increasing from $0°$ to $90°$ .

Hence, the sine function covers its minimum value and maximum value in the given interval.

c.

To determine

Discuss the behaviour of the cosine function for $\theta$ in the range from to $0°$ to $90°$ .

Answer to Problem 87E

The cosine function covers its minimum value and maximum value in the given interval.

Explanation of Solution

Given information:

Discuss the behaviour of the cosine function for $\theta$ in the range from to $0°$ to $90°$ .

Calculation:

From the table, it can be seen that cosine function is decreasing from $0°$ to $90°$ .

Hence, the cosine function covers its minimum value and maximum value in the given interval.

d.

To determine

Use the definitions of the sine and cosine functions to explain the results of parts (b) and (c).

Answer to Problem 87E

The ration of opposite side of hepotenuse increase and so it can be deduced that sine function is increasing in the interval $0°$ to $90°$ .

The ration of adjacent side to hypotenuse decreases and so it can be deduced that cosine function is decreasing in the interval $0°$ to $90°$ .

Explanation of Solution

Given information:

Use the definitions of the sine and cosine functions to explain the results of parts (b) and (c).

Calculation:

Definition of sine function states that it is the ratio of opposite side and hypotenuse . As angle varies from $0°$ to $90°$ , length of opposite side increases while keeping hypotenuse constant. Hence the ration of opposite side of hepotenuse increase and so it can be deduced that sine function is increasing in the interval $0°$ to $90°$ .

Definition of cosine function states that it is the ratio of adjacent side and hypotenuse. As angle varies from $0°$ to $90°$ , length of adjacent side decrease while keeping hypotenuse constant. Hence the ration of adjacent side to hypotenuse decreases and so it can be deduced that cosine function is decreasing in the interval $0°$ to $90°$ .