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Fig 129
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Fig 130
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EXAMPLE 117
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Several angles are sketched in Fig 131, and both their radian and degree measures given
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If is an angle, let (x, y) be the coordinates of the terminal point of the corresponding radius (called the terminal radius) on the unit circle We call P = (x, y) the terminal point corresponding to Look at Fig 132 The number y is called the sine of and is written sin The number x is called the cosine of and is written cos Since (cos , sin ) are coordinates of a point on the unit circle, the following two fundamental properties are immediate: (1) For any number , (sin )2 + (cos )2 = 1
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_ p = _180
x _ 3p/4 = _135
Fig 131
unit circle
P = (x, y)
G cos G
sin G x
Fig 132
For any number , 1 cos 1 and 1 sin 1
Math Note: It is common to write sin2 to mean (sin )2 and cos2 to mean (cos )2 EXAMPLE 118
Compute the sine and cosine of /3
Basics
SOLUTION We sketch the terminal radius and associated triangle (see Fig 133) This is a 30 60 90 triangle whose sides have ratios 1 : 3 : 2 Thus 1 =2 x Likewise, y = 3 x It follows that or y = 3x = or 1 x= 2 3 2
3 sin = 3 2 1 = 3 2
and cos
unit circle 3 2 x
Fig 133
You Try It: The cosine of a certain angle is 2/3 The angle lies in the fourth quadrant What is the sine of the angle Math Note: Notice that if is an angle then and + 2 have the same terminal radius and the same terminal point (for adding 2 just adds one more trip around the circle look at Fig 134) As a result, sin = x = sin( + 2 )
CHAPTER 1 Basics
G cos G
sin G x
unit circle
Fig 134
and cos = y = cos( + 2 ) We say that the sine and cosine functions have period 2 : the functions repeat themselves every 2 units In practice, when we calculate the trigonometric functions of an angle , we reduce it by multiples of 2 so that we can consider an equivalent angle , called the associated principal angle, satisfying 0 < 2 For instance, 15 /2 has associated principal angle 3 /2 (since 15 /2 3 /2 = 3 2 ) and 10 /3 2 /3 has associated principal angle (since 10 /3 2 /3 = 12 /3 = 2 2 )
You Try It: What are the principal angles associated with 7 , 11 /2, 8 /3, 14 /5, 16 /7 What does the concept of angle and sine and cosine that we have presented here have to do with the classical notion using triangles Notice that any angle such that 0 < /2 has associated to it a right triangle in the rst quadrant, with vertex on the unit circle, such that the base is the segment connecting (0, 0) to (x, 0) and the height is the segment connecting (x, 0) to (x, y) See Fig 135
y opposite side
G adjacent side
Basics
unit circle
(x, y)
Fig 135
Then sin = y = and cos = x = adjacent side of triangle x = 1 hypotenuse opposite side of triangle y = 1 hypotenuse
Thus, for angles between 0 and /2, the new de nition of sine and cosine using the unit circle is clearly equivalent to the classical de nition using adjacent and opposite sides and the hypotenuse For other angles , the classical approach is to reduce to this special case by subtracting multiples of /2 Our approach using the unit circle is considerably clearer because it makes the signatures of sine and cosine obvious Besides sine and cosine, there are four other trigonometric functions: y x x cot = y 1 sec = x 1 csc = y tan = sin cos cos = sin 1 = cos 1 = sin =
Whereas sine and cosine have domain the entire real line, we notice that tan and sec are unde ned at odd multiples of /2 (because cosine will vanish there) and cot and csc are unde ned at even multiples of /2 (because sine will vanish there) The graphs of the six trigonometric functions are shown in Fig 136 EXAMPLE 119
Compute all the trigonometric functions for the angle = 11 /4
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