To find sin(A+B) and cos(A-B), we need the values of cos(A) and sin(B). We can find these using the Pythagorean identity sin²(θ) + cos²(θ) = 1. Assuming angles A and B are acute, cos(A) and sin(B) will be positive.

To find sin(A+B) and cos(A-B), we need the values of cos(A) and sin(B). We can find these using the Pythagorean identity sin²(θ) + cos²(θ) = 1. Assuming angles A and B are acute, cos(A) and sin(B) will be positive.

If sin(A) = 3/5 and cos(B) = 12/13, find sin(A+B) and cos(A-B).

Answer: $\sin(A+B) = \frac{56}{65} and \cos(A-B) = \frac{63}{65}$

Step-by-step solution

Step 1: Find cos(A) and sin(B)

To find $\sin(A+B)$ and $\cos(A-B)$ , we need the values of $\cos(A)$ and $\sin(B)$ . We can find these using the Pythagorean identity $\sin^2(\theta) + \cos^2(\theta) = 1$ . Assuming angles A and B are acute, $\cos(A)$ and $\sin(B)$ will be positive.

Step 2: Calculate cos(A) and sin(B)

Substitute the given values of $\sin(A)$ and $\cos(B)$ into the Pythagorean identities. Calculate the squares, subtract from 1, and then take the square root to find $\cos(A)$ and $\sin(B)$ .

Step 3: Apply the sum formula for sin(A+B)

Now we use the sum formula for sine, which states that $\sin(A+B) = \sin(A)\cos(B) + \cos(A)\sin(B)$ . We have all the necessary values to substitute into this formula.

Step 4: Calculate sin(A+B)

Substitute the calculated values of $\sin(A)$ $, \cos(B)$ , $\cos(A)$ and $\sin(B)$ into the sum formula for sine and perform the multiplication and addition.

Step 5: Apply the difference formula for cos(A-B)

Next, we use the difference formula for cosine, which states that $\cos(A-B) = \cos(A)\cos(B) + \sin(A)\sin(B)$ . We will substitute our known values into this formula.

Step 6: Calculate cos(A-B)

Substitute the calculated values of $\cos(A)$ $, \cos(B)$ , $\sin(A)$ and $\sin(B)$ into the difference formula for cosine and perform the multiplication and addition.

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