tan (u-v)

given sin u=3/4 and cos v= -5/13 with U and V in quadrant 2

Use the identity
tan (u-v) = (tanu tanv)/[1-tanu tanv)
Get the values of tanu and tanv from the sines and cosines that you are given. For example, if cos v = -5/13, sin v = 12/13 and tan v = -12/5 (in the second quadrant).

To find the value of tan(u-v) using the given information, follow these steps:

Step 1: Find the values of tan(u) and tan(v) using the given information.
- Since sin(u) = 3/4, we can use the Pythagorean identity to find cos(u): cos(u) = √(1 - sin^2(u)) = √(1 - (3/4)^2) = √(1 - 9/16) = √(7/16). Since u is in the second quadrant, cos(u) is negative. Therefore, cos(u) = -√(7/16).
- To find tan(u), divide sin(u) by cos(u): tan(u) = sin(u) / cos(u) = (3/4) / (-√(7/16)).

- Since cos(v) = -5/13, we can use the Pythagorean identity to find sin(v): sin(v) = √(1 - cos^2(v)) = √(1 - (-5/13)^2) = √(1 - 25/169) = √(144/169) = 12/13. Since v is in the second quadrant, sin(v) is positive.
- To find tan(v), divide sin(v) by cos(v): tan(v) = sin(v) / cos(v) = (12/13) / (-5/13) = -12/5.

Step 2: Substitute the values of tan(u) and tan(v) into the formula for tan(u-v).
tan(u-v) = (tan(u) * tan(v)) / (1 - tan(u) * tan(v))
= ((3/4) / (-√(7/16)) * (-12/5)) / (1 - (3/4) / (-√(7/16)) * (-12/5))

Simplify the expression:

Step 3: Multiply the numerators and denominators.
tan(u-v) = (3/4 * -12/5) / ((-√(7/16)) * (1 - (3/4) * -12/5))
= -36/20 / ((-√(7/16)) * (1 + 36/20))

Step 4: Simplify further if possible.
tan(u-v) = -9/5 / ((-√(7/16)) * (20 + 36)/20)
= -9/5 / ((-√(7/16)) * 56/20)
= -9/5 / (-√(7/16) * 14/5)
= -9/5 / (-√(7)/2 * (2/5))
= -9/5 / (-√(7)/2)

Step 5: Multiply the numerator and denominator by 2 to get rid of the fraction in the denominator.
tan(u-v) = -9/5 * (2/-√(7))
= -18/-5√(7)
= 18/5√(7)

Therefore, tan(u-v) is equal to 18/5√(7).