SOLUTION: Find the exact value of the trigonometric function given that sin u = -12/13 and cos v = 24/25. (Both u and v are in Quadrant IV) a)sin (u+v) b)cos(v-u) c)tan(u-v)

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Question 1179362: Find the exact value of the trigonometric function given that sin u = -12/13 and cos v = 24/25. (Both u and v are in Quadrant IV) a)sin (u+v) b)cos(v-u) c)tan(u-v)
Found 2 solutions by MathLover1, MathTherapy:
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

Find the exact value of the trigonometric function given that
sin+%28u%29+=+-12%2F13 and cos+%28v+%29=+24%2F25

using Pythagorean identity to find cos%28u%29

sin%5E2+%28u%29+%2Bcos%5E2%28u%29=1
%28+-12%2F13%29%5E2+%2Bcos%5E2%28u%29=1
cos%5E2%28u%29=1-%28+-12%2F13%29%5E2
cos%5E2%28u%29=1-144%2F169
cos%5E2%28u%29=25%2F169
cos%28u%29=sqrt%2825%2F169%29
cos%28u%29=5%2F13 or cos%28u%29=+-5%2F13+

In the fourth quadrant, the values for cos are positive only, so use cos%28u%29=+5%2F13+


now find sin%28v%29

sin%5E2+%28v%29+%2Bcos%5E2%28v%29=1
sin%5E2+%28v%29+%2B%2824%2F25%29%5E2=1
sin%5E2+%28v%29+=1-%2824%2F25%29%5E2
sin%5E2+%28v%29+=1-576%2F625
sin%5E2+%28v%29+=49%2F625
sin+%28v%29+=sqrt%2849%2F625%29
sin+%28v%29+=7%2F25 or sin+%28v%29+=-7%2F25

since in Quadrant IV , use sin+%28v%29+=-7%2F25


so we have:
sin+%28u%29+=+-12%2F13 and cos%28u%29=+5%2F13+
sin+%28v%29+=-7%2F25 andcos+%28v+%29=+24%2F25


a)
sin+%28u%2Bv%29+=sin%28u%29+cos%28v%29+%2B+cos%28u%29+sin%28v%29......substitute given values
sin+%28u%2Bv%29+=%28-12%2F13+%29+%2824%2F25%29+%2B+%285%2F13+%29+%28-7%2F25%29
sin+%28u%2Bv%29+=-288%2F325++-7%2F65
sin+%28u%2Bv%29+=-323%2F325


b)
sin+%28u%29+=+-12%2F13 and cos%28u%29=+5%2F13+
sin+%28v%29+=-7%2F25 andcos+%28v+%29=+24%2F25

cos%28v-u%29+=sin%28u%29+sin%28v%29+%2B+cos%28u%29+cos%28v%29
cos%28v-u%29+=%28-12%2F13%29+%28-7%2F25%29+%2B+%285%2F13%29+%2824%2F25%29
cos%28v-u%29+=84%2F325+%2B+24%2F65
cos%28v-u%29+=204%2F325


c)
sin+%28u%29+=+-12%2F13 and cos%28u%29=+5%2F13+
sin+%28v%29+=-7%2F25 andcos+%28v+%29=+24%2F25

tan%28u-v%29=sin%28u-v%29%2Fcos%28u-v%29


tan%28u-v%29=%28-253%2F325%29+%2F+%28204%2F325%29
tan%28u-v%29=-253%2F204

Answer by MathTherapy(10551) About Me  (Show Source):
You can put this solution on YOUR website!

Find the exact value of the trigonometric function given that sin u = -12/13 and cos v = 24/25. (Both u and v are in Quadrant IV) a)sin (u+v) b)cos(v-u) c)tan(u-v)
. With y = - 12, and r = 13, x = 5, since this represents a 5-12-13 PYTHAG. Triple. We also get: 

. With x = 24, and r = 25, y = - 7, since this represents a 7-24-25 PYTHAG. triple. We also get: 

a)      b)  

                                               c) 

                                                   matrix%281%2C3%2C+tan+%28u+-+v%29%2C+%22=%22%2C+%28-+288+%2B+35%29%2F5%2824%29%29 รท %28120+%2B+84%29%2F5%2824%29

                                                   

                                                    highlight_green%28matrix%281%2C3%2C+tan+%28u+-+v%29%2C+%22=%22%2C+%28-+253%29%2F204%29%29