This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here are the evaluations for the trigonometric expressions: First, we need to find the missing trigonometric values for angles A and B using the Pythagorean identity ^2 + ^2 = 1. We assume A and B are acute angles, so their sine and cosine values are positive. Given: A = (12)/(13) and B = (3)/(5). Step 1: Find A. ^2 A = 1 - ^2 A ^2 A = 1 - ((12)/(13))^2 ^2 A = 1 - (144)/(169) ^2 A = (169 - 144)/(169) ^2 A = (25)/(169) A = sqrt((25)/(169)) = (5)/(13) Step 2: Find B. ^2 B = 1 - ^2 B ^2 B = 1 - ((3)/(5))^2 ^2 B = 1 - (9)/(25) ^2 B = (25 - 9)/(25) ^2 B = (16)/(25) B = sqrt((16)/(25)) = (4)/(5) Now we have all the necessary values: A = (12)/(13) A = (5)/(13) B = (4)/(5) B = (3)/(5) a) Evaluate (A + B). Step 3: Use the sum formula for sine, (A + B) = A B + A B. (A + B) = ((12)/(13))((3)/(5)) + ((5)/(13))((4)/(5)) (A + B) = (36)/(65) + (20)/(65) (A + B) = (36 + 20)/(65) (A + B) = (56)/(65) b) Evaluate (A + B). Step 4: Use the sum formula for cosine, (A + B) = A B - A B. (A + B) = ((5)/(13))((3)/(5)) - ((12)/(13))((4)/(5)) (A + B) = (15)/(65) - (48)/(65) (A + B) = (15 - 48)/(65) (A + B) = -(33)/(65) c) Evaluate (2B). Step 5: Use the double angle formula for sine, (2B) = 2 B B. (2B) = 2 ((4)/(5))((3)/(5)) (2B) = 2 ((12)/(25)) (2B) = (24)/(25) What's next?