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MP OM AQ OA AQ
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cot u sec u csc u
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BR OQ OR
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The segments MP, OM, AQ, etc., are directed line segments. The magnitude of a function is given by the length of the corresponding segment, and the sign is given by the indicated direction. The directed segments OQ and OR are to be considered positive when measured on the terminal side of the angle and negative when measured on the terminal side extended.
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7.2 Variations of Trigonometric Functions
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Let P move counterclockwise about the unit circle, starting at A, so that from 0 to 360 . Using Fig. 7.1, see how the trigonometric functions vary (I. / AOP varies continuously increases, D. decreases):
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As Increases from sin cos tan
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0 to 90 I. from 0 to 1 D. from 1 to 0 I. from 0 without limit (0 to ) D. from large positive values to 0( to 0) I. from 1 without limit (1 to )
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90 to 180 D. from 1 to 0 D. from 0 to 1
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180 to 270 D. from 0 to I. from 1 I. from
270 to 360 1 to 0
1 to 0
I. from 0 to 1 I. from large negative values to 0 ( to 0) D. from 0 without limit (0 to )
I. from large negative values to 0 ( to 0) D. from 0 without limit (0 to )
I. from 0 without limit (0 to ) D. from large positive values to 0 ( to 0) D. from 1 without limit ( 1 to ) I. from large negative values to 1( to 1)
I. from large negative values to 1 ( to 1) I. from 1 without limit (1 to )
D. from large positive values to 1 ( to 1)
D. from large positive values to 1( to 1)
D. from 1 without limit ( 1 to )
7.3 Graphs of Trigonometric Functions
In the table on page 78, values of the angle x are given in radians. Whenever a trigonometric function is undefined for the value of x, is recorded instead of a function value. The graphs of the trigonomic functions are shown in Fig. 7.2 on page 76.
CHAPTER 7 Graphs of the Trigonometric Functions
Fig. 7.2
7.4 Horizontal and Vertical Shifts
The graph of a trigonometric function can be shifted vertically by adding a nonzero constant to the function and horizontally by adding a nonzero constant to the angle of the trigonometric function. Figure 7.3(a) is the graph of y sin x and the remaining parts of Fig. 7.3 are the results of shifting this graph. If c is a positive number, then adding it to a trigonometric function results in the graph being shifted up c units [see Fig. 7.3(b)], and subtracting it from a trigonometric function results in the graph being shifted down c units [see Fig. 7.3(c)]. For a positive number d, a trigonometric function is shifted left d units when d is added to the angle [see Fig. 7.3(d)] and shifted right d units when d is subtracted from the angle [see Fig. 7.3(e)].
CHAPTER 7 Graphs of the Trigonometric Functions
x 0 /6 /4 /3 /2 2 /3 3 /4 5 /6
sin x
cos x 1.00 0.87 0.71 0.50 0 0.50 0.71 0.87 1.00 0.87 0.71 0.50 0 0.50 0.71 0.87 1.00
tan x
cot x
sec x 1.00
csc x
0.50 0.71 0.87 1.00 0.87 0.71 0.50 0
0.58 1.00 1.73
1.73 1.00 0.58 0
1.15 1.41 2.00
2.00 1.41 1.15 1.00
1.73 1.00 0.58 0 0.58 1.00 1.73
0.58 1.00 1.73
2.00 1.41 1.15 1.00
1.15 1.41 2.00
7 /6 5 /4 4 /3 3 /2 5 /3 7 /4 11 /6 2
0.50 0.71 0.87 1.00 0.87 0.71 0.50 0
1.73 1.00 0.58 0
1.15 1.41 2.00
2.00 1.41 1.15 1.00
1.73 1.00 0.58 0
0.58 1.00 1.73
2.00 1.41 1.15 1.00
1.15 1.41 2.00
7.5 Periodic Functions
Any function of a variable x, f(x), which repeats its values in definite cycles is called periodic. The smallest range of values of x which corresponds to a complete cycle of values of the function is called the period of the function. It is evident from the graphs of the trigonometric functions that the sine, cosine, secant, and cosecant are of period 2 , while the tangent and cotangent are of period .
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