Question 871906
If csc u=-13/5, P(u) is in Quadrant III and cos v=7/25, P(v) is in Quadrant IV find sec(u-v) and csc(u+v)
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Reference angle (u) is in quadrant III in which sin<0, cos<0.
csc u=-13/5 (working with (5-12-13) reference right triangle in quadrant III in which sin<0, cos<0.
sin u=-5/13
cos u=-12/13
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Reference angle (v) is in quadrant IV in which sin<0, cos>0.
cos v=7/25
sin v=-&#8730;1-cos^2(v)=-&#8730;(1-49/625)=-&#8730;(576/625)=-24/25
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sin(u+v)=sin u*cos v+cos u*sin v=-5/13*7/25+-12/13*-24/25=-35/325+288/325=253/325
csc(u+v)=1/sin(u+v)=325/265
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cos(u-v)=cos u*cos v+sin u*sin v=-12/13*7/25+-5/13*-24/25=-84/325+120/325=36/325
sec(u-v)=1/cos(u-v)=325/36
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Calculator check:
sin u=-5/13 (in quadrant III)
u&#8776;202.62&#730;
cos v=7/25  (in quadrant IV)
v&#8776;286.26&#730;
u+v&#8776;488.88&#730;
u-v&#8776;-83.64&#730;
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sin(u+v)=sin(488.88)&#8776;0.7784…
exact value as calculated=253/325&#8776;0.7784…
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cos(u-v)=cos(-83.64)&#8776;0.1107…
exact value  as calculated=36/325&#8776;0.1107…