Question 888609
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=&#8730;(1-cos^2y)=&#8730;(1-25/36)=&#8730;(11/36)&#8730;11/6
sin(x+y)=sinx*cosy+cosx*siny=3/5*5/6-4/5*&#8730;11/6=15/30-4&#8730;11/30=(15-4&#8730;11)/30
Check:
tanx=-3/4(in quadrant II)
x&#8776;143.13&#730;
cosy=5/6(in quadrant I)
y&#8776;33.56&#730;
(x+y)&#8776;143.13+33.56&#8776;176.69&#730;
sin(x+y)&#8776;sin(176.69)&#8776;0.0577…
exact value=(15-4&#8730;11/30&#8776;0.0577…