Chapter 5.3, Problem 2CFU

To determine

To show: That the value of $\sin \theta$ increases as $\theta$ goes from $0^{\circ }{\text{ to 90}}^{\circ }$ and then decreases as $\theta$ goes from ${90}^{\circ }{\text{ to 180}}^{\circ }$ .

Explanation of Solution

Given information: $\sin \theta$

Calculation:

  

Sine is a cyclic function. As seen in the graph. As you open an angle up from a $0^{\circ }$ angle the ratio of the opposite side over the hypotenuse increases as the opposite side gets bigger.

The maximum you can get is the opposite equals the hypotenuse.

At this point the ratio for $\sin =1$

This happens at $\theta ={90}^{\circ }$ . Past that the opposite side starts shrinking as the angle approaches the x axis again.

  

This is a unit circle. If you are not familiar with it, you soon will be.

In a unit circle we have made the radius 1 for convenience sake.

Since all our trig functions are ratios they do not depend on what we choose as a radius but one makes it easiest.

This is our hypotenuse for any triangle we draw from a point on the unit to the x axis to the origin.

As a point moves from $\left(1,0\right)$ which is on the x-axis around the circle the lengths of x and y constantly change so the values the six trig ratios change.

Sine and cosine cycle up and down from − 1 to 1. The other four have asymptotes due to the fact that they have sine or cosine as denominators and these (sine, cosine) are sometimes equal to 0.

Sine and cosine always have the hypotenuse as a denominator and that is always the radius of the circle and is never equal to 0 and in the unit circle is always 1.

So the sine and cosine graphs are continuous.