GRAPHS OF SINE AND COSINE

LESSON READ-THROUGH
by Dr. Carol JVF Burns (website creator)
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The graphs of the sine and cosine functions are shown below.


graph of $y = \sin x$

graph of $y = \cos x$

Where do these graphs come from?

The sine function gives the $y$-values of points on the unit circle.
The cosine function gives the $x$-values of points on the unit circle.

Since the unit circle has radius $\,1\,$, all its points have coordinates between $\,-1\,$ and $\,1\,$.
That's why both graphs (sine and cosine) are trapped between $\,y = -1\,$ and $\,y = 1\,$.

Here's a way you can visualize the graph of the sine function:

Put your finger at the point $\,(1,0)\,$ on the unit circle.
Twirl it around the circle counter-clockwise (start UP).

The sine function tracks your finger's up/down motions:


A higher level of understanding for the sine function:

In Radian Measure, we ‘wrap’ the real number line around the unit circle.
In this way, every real number is associated with a point (called the terminal point) and a corresponding angle on the unit circle.
The real number is then the radian measure of this angle!

Wrap the real number line
around the unit circle!



Thus, every real number $\,x\,$ ...

... is associated with a point on the unit circle ...

... and a corresponding angle.



In The Unit Circle Approach to Trigonometry, we saw that $\,\sin(x)\,$ is the $y$-value of the terminal point:
As $\,x\,$ goes from $\,0\,$ to $\,\frac{\pi}{2}\,$,
$\,\sin x\,$ goes from $\,0\,$ to $\,1\,$.

(Remember:
$\,\frac{\pi}{2}\,$ radians is $\,90^\circ\,$)
As $\,x\,$ goes from $\,\frac{\pi}{2}\,$ to $\,\pi\,$,
$\,\sin x\,$ goes from $\,1\,$ back to $\,0\,$.

(Remember:
$\,\pi\,$ radians is $\,180^\circ\,$)

The cosine function:

The entire discussion can be repeated to understand the cosine—except that it gives the $x$-values of the points, not the $y$-values!
Here's how one of the graphics above would be adjusted to focus attention on the $x$-value:

Important Characteristics of the Graphs

Properties of both sine and cosine:

Two Trigonometric Identities

Take the sine curve and shift it $\,\frac{\pi}{2}\,$ units to the left—it turns into the cosine curve: $$ \cssId{s50}{\sin(x + \frac{\pi}{2}) = \cos x} $$ Take the cosine curve and shift it $\,\frac{\pi}{2}\,$ units to the right—it turns into the sine curve: $$ \cssId{s52}{\cos(x - \frac{\pi}{2}) = \sin x} $$

Thus, we have two new trigonometric identities!
For all real numbers $\,x\,$:

Master the ideas from this section
by practicing the exercise at the bottom of this page.


When you're done practicing, move on to:
Graphing Generalized Sines and Cosines,
like $\,y=a \sin k(x±b)\,$ and $\,y = a \cos(kx±B)$
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
PROBLEM TYPES:
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AVAILABLE MASTERED IN PROGRESS

(MAX is 11; there are 11 different problem types.)
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