Question

Find the exact value of cos(u+v), sin u=8/17 and cos v =-60/61

Answer 1

The exact value of cos(u+v) is -568/10497.

Step-by-step explanation:

To find the exact value of cos(u+v), we need to use the given information: sin(u) = 8/17 and cos(v) = -60/61. First, we'll find the value of sin(v) using the Pythagorean identity sin^2(v) + cos^2(v) = 1. Solving for sin(v), we get sin(v) = sqrt(1 - cos^2(v)) = sqrt(1 - (-60/61)^2) = sqrt(1 - 3600/3721) = sqrt(121/3721) = 11/61. Now, we can use the sum formula for cosine: cos(u+v) = cos(u)cos(v) - sin(u)sin(v) = (8/17)(-60/61) - (11/61)(8/17) = -480/10497 - 88/10497 = -568/10497. Therefore, the exact value of cos(u+v) is -568/10497.

Answer 2

Assumint that u is in the 1st quadrant and v is in the second quadrant, then given that


`\sin u= (8)/(17) \\ \\ \Rightarrow\cos u= ( √(17^2-8^2) )/(17) \\ \\ = ( √(289-64) )/(17) = ( √(225) )/(17) = (15)/(17)`

Given that


`\cos v=- (60)/(61) \\ \\ \Rightarrow\sin v= ( √(61^2-60^2) )/(61) \\ \\ = ( √(3,721-3,600) )/(61) = ( √(121) )/(61) \\ \\ = (11)/(61)`


`\cos(u+v)=\cos u\cos v-\sin u\sin v \\ \\ = (15)/(17) * \left(-(60)/(61)\right) - (8)/(17) * (11)/(61) \\ \\ =- (900)/(1,037) - (88)/(1,037) = -(988)/(1,037)`