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
Step 1: Simplify the complex number (1+2i)/(2+i). To simplify, multiply the numerator and denominator by the conjugate of the denominator. The conjugate of 2+i is 2-i. (1+2i)/(2+i) = (1+2i)/(2+i) × (2-i)/(2-i) = ((1)(2) + (1)(-i) + (2i)(2) + (2i)(-i))/((2)(2) + (2)(-i) + (i)(2) + (i)(-i)) = (2 - i + 4i - 2i^2)/(4 - 2i + 2i - i^2) Substitute i^2 = -1: = (2 + 3i - 2(-1))/(4 - (-1)) = (2 + 3i + 2)/(4 + 1) = (4 + 3i)/(5) = (4)/(5) + (3)/(5)i Step 2: Find the modulus r of the complex number. The complex number is in the form x + yi, where x = (4)/(5) and y = (3)/(5). The modulus r is given by the formula r = sqrt(x^2 + y^2). r = sqrt(((4)/(5))^2 + ((3)/(5))^2) r = sqrt((16)/(25) + (9)/(25)) r = sqrt((16+9)/(25)) r = sqrt((25)/(25)) r = sqrt(1) r = 1 Step 3: Compare with the given polar form. The problem states that (1+2i)/(2+i) = r( + i ). From our calculation, we found that (1+2i)/(2+i) = 1( + i ). By comparing the two expressions, we can see that r = 1. The final answer is c. 1. What's next?