Trigonometric Functions

1. OPEN ENDED Write an equation giving the value of the Cosine function for an angle measure in its domain. Then, write your equation in the form of an inverse function.
2. OPEN ENDED Write the equation of a trigonometric function with a phase shift of -45°. Then graph the function, and its parent graph.
3. Writing in Math If the formula for the temperature T in degrees Fahrenheit of a city t months into the year is given by T = 50 + 25 sin(6t), explain how to find the average temperature and the maximum and minimum predicted over the year.
4. Writing in Math Use the information on page 842 to explain how you can predict the behavior of tides. Explain why certain tidal characteristics follow the patterns seen in the graph of the sine function.

GET READY for the Lesson
High Tide
Period
Tidal
Range
The rise and fall of tides can have
great impact on the communities
and ecosystems that depend
upon them. One type of tide is a
semidiurnal tide. This means that
bodies of water, like the Atlantic
Ocean, have two high tides and
two low tides a day. Because
tides are periodic, they behave
the same way each day.
Still Water
Level
Low Tide
Graph Trigonometric Functions The diagram below illustrates the water
level as a function of time for a body of water with semidiurnal tides.
High Tide
Water
Level
+
14 16
++++
20 22
24 Time
पिपराजमान
10 12
Low Tide
In each cycle of high and low tides, the pattern repeats itself. Recall that
a function whose graph repeats a basic pattern is said to be periodic.
To find the period, start from any point on the graph and proceed to the
right until the pattern begins to repeat. The simplest approach is to begin
at the origin. Notice that after about 12 hours the graph begins to repeat.
Thus, the period of the function is about 12 hours.
To graph the functions y = sin 0, y = cos 0, or y = tan 0, use values of 0
expressed either in degrees or radians. Ordered pairs for points on these
graphs are of the form (e, sin 6), (0, cos 6), and (0, tan 6), respectively,
45°
60°
90
120°
135
150°
180°
210°
225
270
330°
360°
✓
V3
2
1
V2
12
240
V3
2
300°
V3
2
11 을
1
1
2
315°
V2
2
V2
2
0
-1
글
0
0.7
0.9
1
0.9
OUP
05
0
-0.5
-0.7
-0.9
-1
-0.9
-0.7
-0.5
0
0
1
3
V2
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2
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2
1
V3
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V2
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1
0
1
2