SOLUTION: If tan theta=-3/4 and cos beta=5/6 what is the out come of sin(theta+beta)?
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Question 888609: If tan theta=-3/4 and cos beta=5/6 what is the out come of sin(theta+beta)?
How do you work this problem out> Answer by lwsshak3(11628) (Show Source):
You can put this solution on YOUR website! If tan theta=-3/4 and cos beta=5/6 what is the out come of sin(theta+beta)?
use x for theta
use y for beta
..
tanx=-3/4 (3-4-5) reference right triangle
cosy=5/6
reference angle x is in either quadrant II or quadrant IV in which tan<0
reference angle y is in either quadrant I or quadrant IV in which cos>0
I will assume the smaller angle in both cases. (Note to student: next time, please specify the quadrant in which the reference angles are in.
Identity: sin(theta+beta)=sin(x+y)=sinx*cosy+cosx*siny
cosx=-4/5
sinx=3/5
cosy=5/6
siny=√(1-cos^2y)=√(1-25/36)=√(11/36)√11/6
sin(x+y)=sinx*cosy+cosx*siny=3/5*5/6-4/5*√11/6=15/30-4√11/30=(15-4√11)/30
Check:
tanx=-3/4(in quadrant II)
x≈143.13˚
cosy=5/6(in quadrant I)
y≈33.56˚
(x+y)≈143.13+33.56≈176.69˚
sin(x+y)≈sin(176.69)≈0.0577…
exact value=(15-4√11/30≈0.0577…
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