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)
Answer by lwsshak3(11628)   (Show Source):
You can put this solution on YOUR website! 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)
***
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
..
Reference angle (v) is in quadrant IV in which sin<0, cos>0.
cos v=7/25
sin v=-√1-cos^2(v)=-√(1-49/625)=-√(576/625)=-24/25
..
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
..
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
..
Calculator check:
sin u=-5/13 (in quadrant III)
u≈202.62˚
cos v=7/25 (in quadrant IV)
v≈286.26˚
u+v≈488.88˚
u-v≈-83.64˚
..
sin(u+v)=sin(488.88)≈0.7784…
exact value as calculated=253/325≈0.7784…
..
cos(u-v)=cos(-83.64)≈0.1107…
exact value as calculated=36/325≈0.1107…
|
|
|
