SOLUTION: 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)

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�

RELATED QUESTIONS
sin u = −5/13 and cos v = −15/17 (Both u and v are in Quadrant III.) what is... (answered by greenestamps,ikleyn)

Find the exact value of the trigonometric expression given that sin u = -7/25 and... (answered by ewatrrr)

Find the exact value of the trigonometric function given that sin u = -12/13 and cos v =... (answered by MathLover1,MathTherapy)

Find the exact value of sin(u+v) given that sin u=-12/13 and cos v=-4/5. Both u and v are (answered by lwsshak3)

Find the exact value of the trigonometric function below given that sin u = -7/25 and cos (answered by solver91311,Tatiana_Stebko,MathTherapy)

Find the exact value of the trigonometric function given that sin u = -3/5 and cos v =... (answered by MathLover1)

Find the exact value of the trigonometric function given that sin u = 5/13 and cos v =... (answered by ikleyn,Edwin McCravy,MathTherapy)

Find the exact value of the trigonometric expression when sin(u) = − 5/13 and cos(v)... (answered by Alan3354)

Find the exact value of the trigonometric function given that sin u = −5/13 and... (answered by solver91311)