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