Question

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 find cos(alpha + theta), we can use the trigonometric identity:

cos(a + b) = cos(a)cos(b) - sin(a)sin(b)

Given that sin(theta) = 3/5 and cos(alpha) = 12/13, we can first find cos(theta) and sin(alpha) using the Pythagorean identity:

cos(theta) = sqrt(1 - sin^2(theta))
cos(theta) = sqrt(1 - (3/5)^2)
cos(theta) = sqrt(1 - 9/25)
cos(theta) = sqrt(16/25)
cos(theta) = 4/5

sin(alpha) = sqrt(1 - cos^2(alpha))
sin(alpha) = sqrt(1 - (12/13)^2)
sin(alpha) = sqrt(1 - 144/169)
sin(alpha) = sqrt(25/169)
sin(alpha) = 5/13

Now we can substitute these values into the trigonometric identity:

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.