SOLUTION: find the exact value of the trigonometric function given that sin u= 8/17 and cos v= -9/47 csc(u-v)

Algebra ->  Trigonometry-basics -> SOLUTION: find the exact value of the trigonometric function given that sin u= 8/17 and cos v= -9/47 csc(u-v)      Log On


   

Question 1074974: find the exact value of the trigonometric function given that sin u= 8/17 and cos v= -9/47
csc(u-v)
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
sin%5E2%28u%29=64%2F289
cos%5E2%28u%29=1-sin%5E2%28u%29
cos%5E2%28u%29=1-64%2F289
cos%5E2%28u%29=289%2F289-64%2F289
cos%5E2%28u%29=225%2F289
cos%28u%29=15%2F17 or cos%28u%29=-15%2F17
.
.
Similarly,
cos%5E2%28v%29=81%2F2209
sin%5E2%28v%29=1-cos%5E2%28v%29
sin%5E2%28v%29=1-81%2F2209
sin%5E2%28v%29=2209%2F2209-81%2F2209
sin%5E2%28v%29=2128%2F2209
sin%28v%29=sqrt%282128%29%2F47 or sin%28v%29=-sqrt%282128%29%2F47
.
.
So then,
csc%28u-v%29=1%2Fsin%28u-v%29
and
sin%28u-v%29=sin%28u%29cos%28v%29-sin%28v%29cos%28u%29
So there are 4 possible solutions.
1. cos%28u%29 positive, sin%28v%29 positive
2. cos%28u%29 positive, sin%28v%29 negative
3. cos%28u%29 negative, sin%28v%29 positive
4. cos%28u%29 negative, sin%28v%29 negative
.
.
.
I'll do case #1, you do the others the same way.
.
.
.
sin%28u-v%29=sin%28u%29cos%28v%29-sin%28v%29cos%28u%29
sin%28u-v%29=%288%2F17%29%28-9%2F47%29-%28sqrt%282128%29%2F47%29%2815%2F17%29
sin%28u-v%29=%28-72%2F799%29-%2815%2F799%29%28sqrt%282128%29%29
sin%28u-v%29=-%2872%2B15sqrt%282128%29%29%2F799
So then,
csc%28u-v%29=-799%2F%2872%2B15sqrt%282128%29%29