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.
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Step 1: Find (z_1)/(z_2). Given z_1 = 3 + 5i and z_2 = 2 + 4i. To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator. The conjugate of z_2 = 2 + 4i is 2 - 4i. (z_1)/(z_2) = (3 + 5i)/(2 + 4i) × (2 - 4i)/(2 - 4i) Multiply the numerators: (3 + 5i)(2 - 4i) = 3(2) + 3(-4i) + 5i(2) + 5i(-4i) = 6 - 12i + 10i - 20i^2 Since i^2 = -1: = 6 - 2i - 20(-1) = 6 - 2i + 20 = 26 - 2i Multiply the denominators: (2 + 4i)(2 - 4i) = 2^2 - (4i)^2 = 4 - 16i^2 Since i^2 = -1: = 4 - 16(-1) = 4 + 16 = 20 Now combine the numerator and denominator: (z_1)/(z_2) = (26 - 2i)/(20) = (26)/(20) - (2)/(20)i (z_1)/(z_2) = (13)/(10) - (1)/(10)i Step 2: Determine the modulus of z_1. Given z_1 = 3 + 5i. The modulus |z_1| is calculated as sqrt(x^2 + y^2). |z_1| = sqrt(3^2 + 5^2) = sqrt(9 + 25) = sqrt(34) Step 3: Determine the argument of z_1. Given z_1 = 3 + 5i. The argument (z_1) is such that = (y)/(x). Since x=3 and y=5 are both positive, z_1 is in the first quadrant. = (5)/(3) = ((5)/(3)) In degrees: ≈ 59.04^ In radians: ≈ 1.03 radians The results for the complex numbers are: (z_1)/(z_2) = (13)/(10) - (1)/(10)i |z_1| = sqrt(34) (z_1) = ((5)/(3)) ≈ 59.04^ or 1.03 radians Step 4: Solve the logarithm equation _4 3^x - 4 _4 3 = 0. Use the logarithm property _b M^p = p _b M: x _4 3 - 4 _4 3 = 0 Factor out _4 3: (x - 4) _4 3 = 0 Since _4 3 ≠ 0 (because 3 ≠ 4^0 = 1), we must have: x - 4 = 0 x = 4 The solution to the logarithm equation is: x = 4 Step 5: Find (A+B) and (A-B). Given A = (4)/(5) and B = (12)/(13). Angles A and B are both acute, meaning they are in the first quadrant. First, find A and B. For angle A: Since A is acute, A > 0. A = sqrt(1 - ^2 A) = sqrt(1 - ((4)/(5))^2) = sqrt(1 - (16)/(25)) = sqrt((25 - 16)/(25)) = sqrt((9)/(25)) = (3)/(5) For angle B: Since B is acute, B > 0. B = sqrt(1 - ^2 B) = sqrt(1 - ((12)/(13))^2) = sqrt(1 - (144)/(169)) = sqrt((169 - 144)/(169)) = sqrt((25)/(169)) = (5)/(13) Now, calculate (A+B) using the formula (A+B) = A B + A B: (A+B) = ((4)/(5))((12)/(13)) + ((3)/(5))((5)/(13)) = (48)/(65) + (15)/(65) = (48 + 15)/(65) = (63)/(65) Finally, calculate (A-B) using the formula (A-B) = A B + A B: (A-B) = ((3)/(5))((12)/(13)) + ((4)/(5))((5)/(13)) = (36)/(65) + (20)/(65) = (36 + 20)/(65) = (56)/(65) The results for the trigonometric expressions are: (A+B) = (63)/(65) (A-B) = (56)/(65)