Feta and alpha are each a cute angles in standard position. SIN feta =3/5 and cos alpha =12/13. Determine cos(alpha and theta ).
To find cos(alpha + theta), we need to use the cosine addition formula:
cos(alpha + theta) = cos(alpha) * cos(theta) - sin(alpha) * sin(theta)
We are given that cos(alpha) = 12/13, but we don't have a value for sin(alpha). Therefore, we need to find sin(alpha).
To find sin(alpha), we can use the Pythagorean identity:
sin^2(alpha) + cos^2(alpha) = 1
Since we know cos(alpha) = 12/13, we can solve for sin(alpha):
sin^2(alpha) + (12/13)^2 = 1
sin^2(alpha) + 144/169 = 1
sin^2(alpha) = 1 - 144/169
sin^2(alpha) = 25/169
Taking the square root of both sides to solve for sin(alpha), we get:
sin(alpha) = √(25/169)
sin(alpha) = 5/13
Now, let's find sin(theta). We are given that sin(feta) = 3/5, so we can use the Pythagorean identity to solve for cos(feta):
sin^2(feta) + cos^2(feta) = 1
(3/5)^2 + cos^2(feta) = 1
9/25 + cos^2(feta) = 1
cos^2(feta) = 1 - 9/25
cos^2(feta) = 16/25
Taking the square root of both sides to solve for cos(feta), we get:
cos(feta) = ± √(16/25)
Since the angle is in the first quadrant (cute angle in standard position), cos(feta) = √(16/25)
cos(feta) = 4/5
Now that we have sin(alpha), cos(alpha), sin(feta), and cos(feta), we can substitute these values into the cosine addition formula to find cos(alpha + theta):
cos(alpha + theta) = cos(alpha) * cos(theta) - sin(alpha) * sin(theta)
cos(alpha + theta) = (12/13) * cos(theta) - (5/13) * sin(theta)
Since we don't have specific values for alpha or theta, we are not able to find the exact value of cos(alpha + theta). However, we can leave it in terms of cos(theta) and sin(theta).