Expert: Ahmed Salami - 11/13/2009

Question
I need help solving this 6 part math problem. The problem is stated as the following:
Find the exact value of the trig function given that sin u = 3/5 and cos v = -5/13 and u and v are in quadrant II.
a)sin (u-v)
b)tan (u-v)
c)cos (u+v)
d)sin 2u
e)cos 2u
f)tan 2u
Your assistance is greatly appreciated. Thank you.

Answer
Hi Edward,
In quadrant II, sin u and sin v are positive but all of cos u, cos v, tan u and tan v are negative.
Also,
sin(A+B) = sinA.cosB + sinB.cosA
sin(A-B) = sinA.cosB - sinB.cosA
cos(A+B) = cosA.cosB - sinA.sinB
cos(A-B) = cosA.cosB + sinA.sinB
tan(A+B) = [tanA + tanB]/[1 - tanA.tanB]
tan(A-B) = [tanA - tanB]/[1 + tanA.tanB]

From sin²x + cos²x = 1
sinx = √(1 - cos²x)
and
cosx = √(1 - sin²x)

Now, if
sinx = 3/5
cosx = √[1 - (3/5)²]
    = √(1 - 9/25)
    = √16/25
    = 4/5
tanx = sinx/cosx
    = 3/4
If
cosx = 5/13
sinx = √[1 - (5/13)²]
    = √(1 - 25/169)
    = √144/169
    = 12/13
tanx = sinx/cosx
    = 12/5

But because u and v are in in quadrant II,
sin u = 3/5
cos u = -4/5
tan u = -3/4

cos v = -5/13
sin v = 12/13
tan v = -12/5

a) sin(u - v) = sin u.cos v - sin v.cos u
             = (3/5)(-5/13) - (12/13)(-4/5)
             = -15/65 - -48/65
             = 33/65

b) tan(u - v) = [tan u - tan v]/[1 + tan u.tan v]
             = [(-3/4) - (-12/5)]/[1 + (-3/4).(-12/5)]
             = [33/20]/[1 + (36/20)]
             = (33/20)/(56/20)
             = 33/56

c) cos(u + v) = cos u.cos v - sin u.sin v
             = (-4/5)(-5/13) - (3/5)(12/13)
             = 20/65 - 36/65
             = -16/65

d) sin 2u = sin (u + u)
         = sin u.cos u + sin u.cos u
         = 2sin u.cos u
         = 2(3/5)(-4/5)
         = -24/25

e) cos 2u = cos (u + u)
         = cos u.cos u - sin u.sin u
         = cos²u - sin²u
         = (-4/5)² - (3/5)²
         = 16/25 - 9/25
         = 7/25

f) tan 2u = tan (u + u)
         = [tan u + tan u]/[1 - tan u.tan u]
         = (2tan u)/[1 - tan²u]
         = 2(-3/4)/[1 - (-3/4)²]
         = (-3/2)/[1 - (9/16)]
         = (-3/2)/(7/16)
         = -24/7
OR
We could have just said
tan 2u = sin 2u / cos 2u
      = (-24/25)/(7/25)
      = -24/7

Regards. And good luck.