Concept explainers
To show: That the value of
Explanation of Solution
Given information:
Calculation:
Sine is a cyclic function. As seen in the graph. As you open an angle up from a
The maximum you can get is the opposite equals the hypotenuse.
At this point the ratio for
This happens at
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
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.
Chapter 5 Solutions
Advanced Mathematical Concepts: Precalculus with Applications, Student Edition
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