Chapter 5.3, Problem 45E

To determine

To find:values of $\theta$ for which expression is undefined.

Answer to Problem 45E

  $\theta \ne {90}^{\circ }k\text{ where }k\text{ is an integer}$

Explanation of Solution

Given information:: If you ignore friction, the amount of time required for a box to slide down an inclined plane is $\sqrt{\frac{2a}{g\sin \theta \cos \theta }},$ where a is the horizontal distance defined by the inclined plane, g is the acceleration due to gravity and $\theta$ is the angle of the inclined plane.

Calculation:

We have the consideration on the possible value of $\theta ,$

The denominator of the fraction cannot equal 0. To the left is the sine and cosine plotted on the same graph.

The only time the product of the two will be = 0 when one of them is 0.

For $\sin \theta$ this happens at $\theta ={180}^{\circ }k$ where k is an integer.

For $\cos \theta$ this happens for $\theta ={90}^{\circ }+{180}^{\circ }k$ where k is an integer.

This amounts to is for $\theta ={90}^{\circ }k$ where k is an integer either sine or cosine will be zero.

So those values aren’t allowed.

  $\theta \ne {90}^{\circ }k\text{ where }k\text{ is an integer}$