Question

Use the cosine of a sum and cosine of a difference identities to find cos(s+t) and cos(s+t). sin s = 3/5 and sin t = -5/13, s in quadrant Il and t in quadrant IV. Find cos(s+t) .

Answer 1

To find cos(s+t), we can use the cosine of a sum identity: cos(s+t) = cos(s)cos(t) - sin(s)sin(t). We substitute the given values to find cos(s) and cos(t), then substitute them in the cosine of a sum identity to find cos(s+t) = -33/65.

Step-by-step explanation:

To find cos(s+t), we can use the cosine of a sum identity: cos(s+t) = cos(s)cos(t) - sin(s)sin(t). We are given that sin s = 3/5 and sin t = -5/13. Since s is in quadrant II and t is in quadrant IV, we know that cos s is negative and cos t is positive.

Using the Pythagorean identity, we can solve for cos s:

1 - 9/25 = 16/25

cos s = -sqrt(16/25) = -4/5

Similarly, we can solve for cos t:

1 - 25/169 = 144/169

cos t = sqrt(144/169) = 12/13

Now we can substitute the values to find cos(s+t):

cos(s+t) = cos(s)cos(t) - sin(s)sin(t) = (-4/5)(12/13) - (3/5)(-5/13) = -48/65 + 15/65 = -33/65