Question

If √2 cos(A−B)=1, 2sin(A+B)= 3, and A and B are acute angles, find the values of A and B.

Answer 1

To solve for A and B, rearrange the given equations and use trigonometric identities to substitute known values.

Step-by-step explanation:

To solve the given equations, let's start by rearranging the first equation to solve for cos(A-B):

√2 cos(A−B) = 1

cos(A−B) = 1/√2

Now, let's rewrite the second equation in terms of cos(A+B):

2sin(A+B) = 3

sin(A+B) = 3/2

Using the trigonometric identity sin(A+B) = sinAcosB + cosAsinB, we can substitute the known values:

3/2 = sinAcosB + cosAsinB

Since A and B are acute angles, the sine and cosine functions are positive. We can use the pythagorean identity sin^2(A) + cos^2(A) = 1 to rewrite the equation as:

(sinA + cosA)(sinB + cosB) = 3/2

Now, we can solve these equations simultaneously:

cos(A-B) = 1/√2

(sinA + cosA)(sinB + cosB) = 3/2

By substituting the known values, we can solve for A and B.