http://www.ck12.org Chapter 5. Trigonometric Functions
Example C
What happens on either side of the sine and cosine graphs? Can you explain why?
Solution: The graphs of the sine (blue) and cosine (red) functions repeat forever in both directions.
If you think about the example with the Ferris wheel, the ride will keep on spinning and has been spinning for-
ever. This is why the same cycle of the graph repeats over and over.
Concept Problem Revisited
The unit circle produces the parent function sine and cosine graphs. When the unit circle is shifted up or down,
made wider or narrower, or spun faster or slower in either direction, then the graphs of the sine and cosine functions
will be transformed using basic function transformation rules.
Vocabulary
Thesinusoidal function familyrefers to either sine or cosine waves since they are the same except for a horizontal
shift. This function family is also called the periodic function familybecause the function repeats after a given
period of time.
Guided Practice
- How are the sine and cosine graphs the same and how are they different?
- Where are two maximums and two minimums of the sine graph?
- In the interval[− 2 π, 4 π)where does cosine have zeroes?
Answers: - The sine graph is the same as the cosine graph offset byπ 2. Besides this shift, both curves are identical due to the
perfect symmetry of circles. - One maximum of the sine graph occurs at(π 2 , 1 ). One minimum occurs at(^32 π,− 1 ). This is one cycle of the
sine graph. Since it completes a cycle every 2π, when you add 2πto anx-coordinate you will be on the same point
of the cycle giving you another maximum or minimum.
( 5 π
2 ,^1
)is another maximum.( 7 π
2 ,−^1)is another minimum.- Observe where the cosine curve hasx-coordinates equal to zero. Note that 4πis excluded from the interval. The
values are−^32 π,−π 2 ,π 2 ,^32 π,^52 π,^72 π.