Chapter 4.4, Problem 102E

To determine

To describe: the changes in the values of x, y, $\sin \theta$ , $\cos \theta$ and $\tan \theta$ as $\theta$ increases continuously from $0°$ to $90°$ .

Answer to Problem 102E

x − decreases

y − increases

  $\sin \theta$ -increases

  $\cos \theta$ -decreases

  $\tan \theta$ -increases

Explanation of Solution

Given:

Consider an angle in standard position with $r=12$ centimeters, as shown in the figure:

Looking at the given figure it can be observed that $\theta$ increases continuously from $0°$ to $90°$ the angle will move closer to the y-axis. So, the value of x will decrease while the value of y will increase.

Now consider, $\sin \theta =\frac{y}{r}$

Here as r is constant but value of y is increasing so, the value of $\sin \theta$ will also increase.

Consider,

  $\cos \theta =\frac{x}{r}$

Here as r is constant but value of x is decreasing so, the value of $\cos \theta$ will also decrease.

Consider,

  $\tan \theta =\frac{y}{x}$

Here as the numerator value y is increasing but the denominator value x is decreasing so, the value of $\tan \theta$ will increase