Question

For positive acute angles A and B, it is known that cosA=28/53 and sinB=12/37. Find the value of cos(A B) in simplest form

Answer 1

To find the value of cos(AB), we need to find the values of cos(A) and cos(B). Given that cosA = 28/53 and sinB = 12/ 37, we can use the Pythagorean identity sin^2(B) + cos^2(B) = 1 to find cos(B). Now that we have cos(A) and cos(B), we can find cos(AB) by using the cosine addition formula.

Step-by-step explanation:

To find the value of cos(AB), we need to find the values of cos(A) and cos(B).

Given that cosA = 28/53 and sinB = 12/ 37, we can use the Pythagorean identity sin^2(B) + cos^2(B) = 1 to find cos(B).

Since sinB = 12/37, we can substitute this value into the identity to get (12/37)^2 + cos^2(B) = 1. Solving for cos(B), we get cos(B) = sqrt(1 - (12/37)^2).

Now that we have cos(A) and cos(B), we can find cos(AB) by using the cosine addition formula cos(AB) = cos(A) * cos(B) - sin(A) * sin(B).

Substituting the given values, we get cos(AB) = (28/53) * sqrt(1 - (12/37)^2) - (sqrt(1 - (28/53)^2) * (12/37)).

Simplifying further will give the final value of cos(AB) in simplest form.