find the exact value

Question

find the exact value of sin(u-v) given that sin u=3/5, and sin v=4/5. with u and v in the first quadrant.

find the exact value of cos(u+v) given that sin u=4/5, and sin v=5/13. with you and v in the second quadrant.

Answer 1

A) If sin u = 3/5, then using the Pythagorean Rule, cos u = 4/5

Likewise, if sin v = 4/5, cos v = 3/5

Using the trig difference formula, sin(u-v) = (sin u)(cos v) - (cos u)(sin v) = (3/5)(3/5) - (4/5)(4/5) = - 7/25

note that sin v is > sin u, thus the negative value for sin (u-v). Since u and v are in the first quadrant, this places (u-v) in Q4

B) Sin u = 4/5, so cos u = - 3/5 (cosine is negative in Q2)

Sin v = 5/13, so cos v = -12/13 " "

Trig sum formula: cos(u+v) = Cos u* cos v = sin u *sin v = (-3/5)(-12/13) - (4/5)(5/13) = 27/65

Cosine is positive, which places (u+v) in Q4

Answer 2

Use the theorem of Pythagoras to get cos u = 3/5 and cos v = 12/13.

Then use the formulae for the sin and cos of the sum and difference of 2 angles.