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
Question 1: If A = (4)/(3) and B = (5)/(13), where both A and B are acute angles, what is the exact value of (A + B)? Step 1: Determine A and A from A = (4)/(3). Since A is an acute angle, we can use a right-angled triangle. Opposite side = 4, Adjacent side = 3. Hypotenuse = sqrt(4^2 + 3^2) = sqrt(16 + 9) = sqrt(25) = 5. Therefore, A = OppositeHypotenuse = (4)/(5) and A = AdjacentHypotenuse = (3)/(5). Step 2: Determine B from B = (5)/(13). Since B is an acute angle, we can use the identity ^2 B + ^2 B = 1. ^2 B + ((5)/(13))^2 = 1 ^2 B + (25)/(169) = 1 ^2 B = 1 - (25)/(169) = (169 - 25)/(169) = (144)/(169) Since B is acute, B is positive. B = sqrt((144)/(169)) = (12)/(13) Step 3: Use the sum formula for sine, (A+B) = A B + A B. Substitute the values found: (A+B) = ((4)/(5))((5)/(13)) + ((3)/(5))((12)/(13)) (A+B) = (20)/(65) + (36)/(65) (A+B) = (20 + 36)/(65) (A+B) = (56)/(65) Comparing this with the given options, the correct option is (b). The exact value of (A + B) is (b) (56)/(65). What's next? Send 'em!