Question 923447
Suppose sin x = 1/7
cos x > 0, sin y = −2/5
and cos y < 0. Then
cos x =
cos y =
Find each of the following quantities:
sin(x + y) =
cos(x + y) =
tan(x + y) 
***
reference angle x is in quadrant I where sin>0, cos>0
sinx=1/7
cosx=&#8730;(1-sin^2x)=&#8730;(1-1/49)=&#8730;(48/49)=&#8730;48/7)
..
reference angle y is in quadrant III where sin<0, cos<0
siny=-2/5
cosy=-&#8730;(1-sin^2y)=-&#8730;(1-4/25)=-&#8730;(21/25)=-&#8730;21/5
..
sin(x+y)=sinxcosy+cosxsiny=1/7*-&#8730;21/5+&#8730;48/7*-2/5=-&#8730;21/35-2&#8730;48/35=-(&#8730;21+2&#8730;48)/35
cos(x+y)=cosxcosy-sinxsiny=&#8730;48/7*-&#8730;21/5-1/7*-2/5=-&#8730;1008/35+2/35=(-&#8730;1008+2)/35
tan(x+2)=sin(x+2)/cos(x+2)=-(&#8730;21+2&#8730;48)/(-&#8730;1008+2)

check:
sinx=1/7
x&#8776;8.21&#730;
siny=-2/5
y&#8776;203.58&#730;
x+y&#8776;203.58+8.21&#8776;211.79
..
sin(x+y)&#8776;sin(211.79)&#8776;-0.5268 (w/calculator)
exact value as computed=-(&#8730;21+2&#8730;48)/35&#8776;-0.5268
..
cos(x+y)&#8776;cos(211.79)&#8776;-0.8500 (w/calculator)
exact value as computed=(-&#8730;1008+2)/35&#8776;-0.8500
..
tan(x+y)=tan(211.79)&#8776;0.6198 (w/calculator)
exact value as computed=-(&#8730;21+2&#8730;48)/(-&#8730;1008+2)&#8776;.5268/.8500&#8776;0.6198