Theta and alpha are each acute angles in standard position. Sin theta=3/5 and cos alpha=12/13. Determine cos(alpha+theta)

Answer 1

To determine cos(alpha+theta), we first need to determine the values of sin(alpha) and cos(theta).

Since sin(theta) = 3/5, we can determine the value of cos(theta) using the Pythagorean identity sin^2(theta) + cos^2(theta) = 1.

(3/5)^2 + cos^2(theta) = 1
9/25 + cos^2(theta) = 1
cos^2(theta) = 16/25
cos(theta) = ±4/5

Since theta is an acute angle, cos(theta) must be positive. So, cos(theta) = 4/5.

Similarly, since cos(alpha) = 12/13, we can determine the value of sin(alpha) using the Pythagorean identity sin^2(alpha) + cos^2(alpha) = 1.

sin^2(alpha) + (12/13)^2 = 1
sin^2(alpha) + 144/169 = 1
sin^2(alpha) = 25/169
sin(alpha) = ±5/13

Since alpha is an acute angle, sin(alpha) must be positive. So, sin(alpha) = 5/13.

Now, we can use the angle addition formula for cosine:
cos(alpha+theta) = cos(alpha)cos(theta) - sin(alpha)sin(theta)
cos(alpha+theta) = (12/13)(4/5) - (5/13)(3/5)
cos(alpha+theta) = 48/65 - 15/65
cos(alpha+theta) = 33/65

Therefore, cos(alpha+theta) = 33/65.