Question

If 180°<α<270°, cos⁡ α=−8/17, 270°<β<360°, and sin⁡ β=−4/5, what is cos⁡(α+β)?

Answer 1

To find cos(α+β), we use trigonometric identities and the given values of cos α and sin β. By applying the identities, we calculate sin(α+β) and cos(α+β).

Step-by-step explanation:

To find cos(α+β), we need to use the trigonometric identities and the given values of cos α and sin β.

Using the trigonometric identity sin(a+b) = sin a cos b + cos a sin b, we can find sin(α+β).

sin(α+β) = sin α cos β + cos α sin β = (-8/17)(√(1 - (-4/5)^2)) + (-4/5)(√(1 - (-8/17)^2)) = -32√833/(17√85).

Next, using the trigonometric identity cos(a+b) = cos a cos b - sin a sin b, we can find cos(α+β).

cos(α+β) = cos α cos β - sin α sin β = (-8/17)(√(1 - (-8/17)^2)) - (-4/5)(√(1 - (-4/5)^2)) = 61√85/(17√833).