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cos a cos b
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CHAPTER 9 Trigonometric Functions of Two Angles
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and are any angles.
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9.2 Show that (1) and (2) of Prob. 9.1 are valid when
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First check the formulas for the case sin (0 and cos (0 0 ) 0 ) 0 and
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0 . Since 0 1 1 1 1 0 0 0 0 1 sin 0 cos 0
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the formulas are valid for this case. Next, it will be shown that if (1) and (2) are valid for any two given angles and , the formulas are also valid when, say, is increased by 90 . Let and be two angles for which (1) and (2) hold and consider (a) sin ( and (b) cos ( 90 ) 90 ) sin ( cos ( 90 ) cos 90 ) cos cos ( sin ( 90 ) sin 90 ) sin
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From the graphs in Sec. 7.3 we see that sin ( 90 ) 90 ) cos ( ) and cos ( 90 ) (a ) cos ( sin ( sin ( ) ) ) cos sin sin cos cos cos
cos and cos ( 90 ) sin . It follows that sin sin ( ). Then (a) and (b) reduce to cos cos sin sin
( sin ) sin cos cos sin sin
and or
(b )
which, by assumption, are valid relations. Thus, (a) and (b) are valid relations. The same argument may be made to show that if (1) and (2) are valid for two angles and , they are also valid when is increased by 90 . Thus, the formulas are valid when both and are increased by 90 . Now any positive angle can be expressed as a multiple of 90 plus , where is either 0 or an acute angle. Thus, by a finite number of repetitions of the argument, we show that the formulas are valid for any two given positive angles. It will be left for the reader to carry through the argument when, instead of an increase, there is a decrease of 90 and thus to show that (1) and (2) are valid when one angle is positive and the other negative, and when both are negative.
9.3 Prove tan (a
tan a tan b . 1 tan a tan b
b) sin (a cos (a b) b) sin a cos b cos a cos b sin a cos b cos a cos b cos a cos b cos a cos b cos a sin b sin a sin b cos a sin b cos a cos b sin a sin b cos a cos b tan a tan b 1 tan a tan b
tan (a
9.4 Prove the subtraction formulas.
sin ( ) sin [ sin cos ( ) cos [ cos tan (a b) ( (cos ) ( (cos ) ( b)] )] )] sin cos cos ( ( sin ) cos ( ) cos sin ) sin ( ) cos sin ( ) sin sin sin
cos sin
cos sin
( sin )
tan [a
tan a tan ( b) 1 tan a tan ( b) 1 tan a tan b tan a tan b
tan a ( tan b) tan a ( tan b)
CHAPTER 9 Trigonometric Functions of Two Angles
9.5 Find the values of the sine, cosine, and tangent of 15 , using (a) 15 60 45 .
(a) sin 15 sin (45 1 # 23 2 30 ) sin 45 cos 30 23 222 22 A 23 4 1 cos 45 sin 30 22 A 23 4 sin 45 sin 30 1B 26 4 1 1 1> 23 1(1> 23) 23 23 1 1 1# 1 2 22 2 23 22 1B 26 4 22
30 and (b) 15
22 cos 15
1 #1 22 2 30 ) 1 #1 22 2 30 ) 1
cos (45 1 # 23 22 2
cos 45 cos 30
tan 15
tan (45
tan 45 tan 30 tan 45 tan 30
(b) sin 15
sin (60 26 4
45 ) 22
sin 60 cos 45
cos 60 sin 45
23 # 1 2 22
22 A 23 4
cos 15
cos (60 26 4
45 ) 22
cos 60 cos 45
sin 60 sin 45
1# 1 2 22
23 # 1 2 22
22 A 23 4
tan 15
tan (60
45 )
tan 60 tan 45 1 tan 60 tan 45
23 23
9.6 Find the values of the sine, cosine, and tangent of /12 radians.
Since /3 and /4 are special angles and /3
sin cos p 12 p 12 p 12 sin Q p 3 p 3 p R 4 p R 4 p R 4 1 1 23 23 sin p p cos 3 4 cos
p p sin 3 4
/12, they can be used to find the values needed.
23 # 22 2 2 1 # 22 2 2 26 4 22 4 22 4 26 4 26 4 22 4 26 22
cos Q tan Q 23 23
23 # 22 p p p p 1 # 22 cos sin sin 3 4 3 4 2 2 2 2 p p tan tan 23 1 23 1 3 4 p p 1 23(1) 1 23 1 tan tan 3 4 cos 1 # 23 1 23 1 1 3 223 3 1 1 4 2 23 2 2
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