Find the magnitude of the complex number \(z = 3 \cos(\theta) + 3i \sin(\theta)\) where \(\theta = \frac{\pi}{4}\).
Answer
Next, we calculate the magnitude of the complex number using the formula \(|z| = \sqrt{Re(z)^2 + Im(z)^2}\), where Re(z) is the real part of z and Im(z) is the imaginary part of z. So, \(|z| = \sqrt{\left(\frac{3\sqrt{2}}{2}\right)^2 + \left(\frac{3\sqrt{2}}{2}\right)^2} = \sqrt{\frac{18}{2} + \frac{18}{2}} = \sqrt{18} = 3\sqrt{2}\).
Step 1 ：First, we substitute \(\theta = \frac{\pi}{4}\) into the complex number, so \(z = 3 \cos\left(\frac{\pi}{4}\right) + 3i \sin\left(\frac{\pi}{4}\right) = 3 \cdot \frac{\sqrt{2}}{2} + 3i \cdot \frac{\sqrt{2}}{2} = \frac{3\sqrt{2}}{2} + \frac{3\sqrt{2}}{2}i\).
Step 2 ：Next, we calculate the magnitude of the complex number using the formula \(|z| = \sqrt{Re(z)^2 + Im(z)^2}\), where Re(z) is the real part of z and Im(z) is the imaginary part of z. So, \(|z| = \sqrt{\left(\frac{3\sqrt{2}}{2}\right)^2 + \left(\frac{3\sqrt{2}}{2}\right)^2} = \sqrt{\frac{18}{2} + \frac{18}{2}} = \sqrt{18} = 3\sqrt{2}\).
