Question 925540
Evaluate cos(a &#8722; b) if cos a = 4/5 tan a < 0, tan b = &#8722;&#8730;15 and cos b < 0
cos(a-b) =
and
Evaluate sin(x + y) if sin x = 3/5 sec x > 0, cos y = &#8722; (2&#8730;5)/5 and tan y < 0.
sin(x + y) = 
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cos(a)=4/5
tan(a)<0
reference angle a is in quadrant IV where cos>0, sin<0
sin(a)=3/5 (working with a (3-4-5) reference right triangle in quadrant IV)
..
tan(b)=-&#8730;15/1
cos(b)>0
reference angle b is in quadrant IV where cos>0, sin<0
hypotenuse of reference right triangle=&#8730;(&#8730;15^2)+1^2)=&#8730;(16)=4
cos(b)=1/4
sin(b)=-&#8730;15/4
..
cos(a-b)=cos(a)cos(b)+sin(a)sin(b)
=4/5*1/4+3/5*&#8730;15/4
=4/20+3&#8730;15/20=4+3&#8730;15/20
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check:
cos(a)=4/5
a&#8776;323.13&#730;
tan(b)=-&#8730;15
b&#8776;284.48
..
a-b&#8776;323.13-284.48&#8776;38.62&#730;
cos(a-b)&#8776;cos(38.65)&#8776;0.7809
exact value as computed=4+3&#8730;15/20&#8776;0.7809
..
sinx=3/5
secx>0
reference angle x is in quadrant I where cos>0, sin>0
cosx=4/5 (working with a (3-4-5) reference right triangle in quadrant I)
..
cosy=-2&#8730;5/5
tany<0
reference angle y is in quadrant II where cos<0, sin>0
siny=&#8730;(1-cos^2(y))=&#8730;(1-20/25)=&#8730;(5/25)=&#8730;5/5
sin(x+y)=sinxcosy+cosxsiny=3/5*-2&#8730;5/5+4/5*&#8730;5/5=-6&#8730;5/25+4&#8730;5/25=-2&#8730;5/25
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check:
sinx=3/5
x=36.87
cosy=-2&#8730;5/5
y=153.43
x+y=36.87+153.43=190.29
sin(x+y)=sin(190.30)&#8776;-0.1788
exact value as computed=-2&#8730;5/25&#8776;-0.1788