Question

Assuming that cos/alpha =3/7 with /alpha between pi/2 and pi and that cos/beta= 4/7 with between 3pi/2 and 2pi ​, evaluate cos (alpha +beta) exactly.

Answer 1

To find the exact value of cos(α + β), first find the sine values of α and β using the Pythagorean identity, then apply the cosine addition formula. In this case, cos(α + β) = (12 + √1320) / 49.

Step-by-step explanation:

To evaluate cos(α + β) exactly, given that cos(α) = 3/7 with α between π/2 and π, and cos(β) = 4/7 with β between 3π/2 and 2π, we need to use the cosine addition formula:

cos(α + β) = cos(α)cos(β) - sin(α)sin(β).

Since α is in the second quadrant, sin(α) will be positive, and since β is in the fourth quadrant, sin(β) will be negative. However, the formula requires the sine values as well, which we can find using the Pythagorean identity:

sin(α) = √(1 - cos2(α))
sin(β) = -√(1 - cos2(β))

Hence, for α:

sin(α) = √(1 - (3/7)2) = √(1 - 9/49) = √(40/49) = √40/7

And for β:

sin(β) = -√(1 - (4/7)2) = -√(1 - 16/49) = -√(33/49) = -√33/7

Plugging these into the cosine addition formula:

cos(α + β) = (3/7)(4/7) - (√40/7)(-√33/7)

cos(α + β) = 12/49 + √(1320)/49

cos(α + β) = (12 + √1320) / 49.