Plot each complex number in polar form and then con- vert to standard form. \( \begin{array}{ll}\text { 10. } 3 \text { cis } \frac{4 \pi}{3} & \text { 11. } 4 \text { cis } \frac{\pi}{6} \\ \text { 12. } 2 \text { cis } \frac{7 \pi}{4} & \text { 13. } 2 \sqrt{2} \text { cis } \frac{3 \pi}{4}\end{array} \)

Sure! Let's convert each given complex number from polar form to standard form \( a + bi \) and describe their positions on the complex plane. ### 10. \( 3 \, \text{cis} \, \frac{4\pi}{3} \) **Conversion to Standard Form:** \[ 3 \, \text{cis} \, \frac{4\pi}{3} = 3 \left( \cos\left(\frac{4\pi}{3}\right) + i \sin\left(\frac{4\pi}{3}\right) \right) \] Calculate the cosine and sine: \[ \cos\left(\frac{4\pi}{3}\right) = -\frac{1}{2}, \quad \sin\left(\frac{4\pi}{3}\right) = -\frac{\sqrt{3}}{2} \] So, \[ 3 \left( -\frac{1}{2} + i \left( -\frac{\sqrt{3}}{2} \right) \right) = -\frac{3}{2} - \frac{3\sqrt{3}}{2}i \] **Standard Form:** \[ - \frac{3}{2} - \frac{3\sqrt{3}}{2}i \] **Position on the Complex Plane:** - **Quadrant:** III (both real and imaginary parts are negative) - **Modulus:** 3 - **Angle:** \( \frac{4\pi}{3} \) radians (240°) --- ### 11. \( 4 \, \text{cis} \, \frac{\pi}{6} \) **Conversion to Standard Form:** \[ 4 \, \text{cis} \, \frac{\pi}{6} = 4 \left( \cos\left(\frac{\pi}{6}\right) + i \sin\left(\frac{\pi}{6}\right) \right) \] Calculate the cosine and sine: \[ \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}, \quad \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \] So, \[ 4 \left( \frac{\sqrt{3}}{2} + i \cdot \frac{1}{2} \right) = 2\sqrt{3} + 2i \] **Standard Form:** \[ 2\sqrt{3} + 2i \] **Position on the Complex Plane:** - **Quadrant:** I (both real and imaginary parts are positive) - **Modulus:** 4 - **Angle:** \( \frac{\pi}{6} \) radians (30°) --- ### 12. \( 2 \, \text{cis} \, \frac{7\pi}{4} \) **Conversion to Standard Form:** \[ 2 \, \text{cis} \, \frac{7\pi}{4} = 2 \left( \cos\left(\frac{7\pi}{4}\right) + i \sin\left(\frac{7\pi}{4}\right) \right) \] Calculate the cosine and sine: \[ \cos\left(\frac{7\pi}{4}\right) = \frac{\sqrt{2}}{2}, \quad \sin\left(\frac{7\pi}{4}\right) = -\frac{\sqrt{2}}{2} \] So, \[ 2 \left( \frac{\sqrt{2}}{2} + i \left( -\frac{\sqrt{2}}{2} \right) \right) = \sqrt{2} - \sqrt{2}i \] **Standard Form:** \[ \sqrt{2} - \sqrt{2}i \] **Position on the Complex Plane:** - **Quadrant:** IV (real part positive, imaginary part negative) - **Modulus:** 2 - **Angle:** \( \frac{7\pi}{4} \) radians (315°) --- ### 13. \( 2\sqrt{2} \, \text{cis} \, \frac{3\pi}{4} \) **Conversion to Standard Form:** \[ 2\sqrt{2} \, \text{cis} \, \frac{3\pi}{4} = 2\sqrt{2} \left( \cos\left(\frac{3\pi}{4}\right) + i \sin\left(\frac{3\pi}{4}\right) \right) \] Calculate the cosine and sine: \[ \cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2}, \quad \sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2} \] So, \[ 2\sqrt{2} \left( -\frac{\sqrt{2}}{2} + i \cdot \frac{\sqrt{2}}{2} \right) = -2 + 2i \] **Standard Form:** \[ -2 + 2i \] **Position on the Complex Plane:** - **Quadrant:** II (real part negative, imaginary part positive) - **Modulus:** \( 2\sqrt{2} \) - **Angle:** \( \frac{3\pi}{4} \) radians (135°) --- ### Summary | Problem | Polar Form | Standard Form | Quadrant | Modulus | Angle (radians) | |---------|---------------------------|-------------------------|----------|---------|-----------------| | 10 | \( 3 \, \text{cis} \, \frac{4\pi}{3} \) | \( -\frac{3}{2} - \frac{3\sqrt{3}}{2}i \) | III | 3 | \( \frac{4\pi}{3} \) | | 11 | \( 4 \, \text{cis} \, \frac{\pi}{6} \) | \( 2\sqrt{3} + 2i \) | I | 4 | \( \frac{\pi}{6} \) | | 12 | \( 2 \, \text{cis} \, \frac{7\pi}{4} \) | \( \sqrt{2} - \sqrt{2}i \) | IV | 2 | \( \frac{7\pi}{4} \) | | 13 | \( 2\sqrt{2} \, \text{cis} \, \frac{3\pi}{4} \) | \( -2 + 2i \) | II | \( 2\sqrt{2} \) | \( \frac{3\pi}{4} \) | These conversions allow you to plot each complex number accurately on the complex plane by identifying their real and imaginary components.