Question

what is the exact value of cos(x+y) if tan=x sqare root 3 over 3 and sin y=square root 2 over 2?

Answer 1

tan x = √3/3 = 1/√3 means the hypotenuse is 2
sin x = 1/2
cos x = √3/2

sin y = √2/2 = 1/√2 means the other leg is 1
cos y = 1/√2

Now just plog and chug:

cos(x+y) = cosx cosy - sinx siny
= √3/2 * 1/√2 - 1/2 * 1/√2
= √3/2√2 - 1/2√2
= (√3-1)/2√2