2 De moiure's theorem. 3 biscuss graph of give polar equations and how normals and tangents are determine using condinates geometry.

Sure, I can help you with these topics. It seems like you’re interested in: 1. **De Moivre's Theorem** 2. **Graphing Polar Equations** 3. **Determining Normals and Tangents Using Coordinate Geometry** Let's delve into each of these topics step by step. --- ## 1. De Moivre's Theorem ### **Statement of De Moivre's Theorem** De Moivre's Theorem is a fundamental formula in complex number theory and trigonometry. It states that for any real number \( \theta \) and any integer \( n \): \[ (\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta) \] ### **Applications of De Moivre's Theorem** 1. **Raising Complex Numbers to Powers:** - Simplifies calculations involving powers of complex numbers in trigonometric form. 2. **Finding Roots of Complex Numbers:** - Facilitates finding \( n \)-th roots of complex numbers by applying the theorem and considering multiple angles. 3. **Deriving Trigonometric Identities:** - Useful in proving various trigonometric identities and relationships. ### **Example** **Compute \((\cos 30° + i \sin 30°)^3\) using De Moivre's Theorem.** **Solution:** Using De Moivre's Theorem: \[ (\cos 30° + i \sin 30°)^3 = \cos(3 \times 30°) + i \sin(3 \times 30°) = \cos 90° + i \sin 90° = 0 + i(1) = i \] --- ## 2. Graphing Polar Equations ### **Understanding Polar Coordinates** In polar coordinates, a point in the plane is determined by: - \( r \): The radial distance from the origin. - \( \theta \): The angle measured from the positive \( x \)-axis. ### **Common Polar Equations and Their Graphs** 1. **Circle:** - **Equation:** \( r = a \) (a circle with radius \( a \) centered at the origin) - **Special Case:** \( r = 2a \cos \theta \) or \( r = 2a \sin \theta \) (circles shifted along the x or y-axis) 2. **Line:** - **Equation:** \( \theta = \alpha \) (a straight line making an angle \( \alpha \) with the positive \( x \)-axis) 3. **Rose Curve:** - **Equation:** \( r = a \cos(k\theta) \) or \( r = a \sin(k\theta) \) - **Petals:** If \( k \) is odd, number of petals = \( k \); if even, number of petals = \( 2k \) 4. **Limaçon:** - **Equation:** \( r = a + b \cos \theta \) or \( r = a + b \sin \theta \) - **Types:** Depending on the ratio of \( a \) and \( b \), can form dimpled, cardioid, or limaçon with an inner loop. 5. **Spiral:** - **Equation:** \( r = a\theta \) (Archimedean spiral) ### **Graphing Steps** 1. **Convert to Cartesian Coordinates (if needed):** - \( x = r \cos \theta \) - \( y = r \sin \theta \) 2. **Plot Key Points:** - Compute \( r \) for various \( \theta \) values. 3. **Determine Symmetry:** - Identify if the graph is symmetric about the \( polar \) axis, the line \( \theta = \frac{\pi}{2} \), or the origin. 4. **Sketch the Graph:** - Connect the plotted points smoothly, considering the symmetry. --- ## 3. Determining Normals and Tangents Using Coordinate Geometry ### **Tangents in Polar Coordinates** To find the equation of the tangent to a polar curve \( r = f(\theta) \) at a given point \( \theta = \theta_0 \): 1. **Find \( \frac{dy}{dx} \):** Using the parametric derivatives: \[ \frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} = \frac{r' \sin \theta + r \cos \theta}{r' \cos \theta - r \sin \theta} \] where \( r' = \frac{dr}{d\theta} \). 2. **Cartesian Coordinates of the Point:** \[ x = r \cos \theta, \quad y = r \sin \theta \] 3. **Equation of the Tangent Line:** \[ y - y_0 = \left( \frac{dy}{dx} \right) (x - x_0) \] ### **Normals in Polar Coordinates** The normal line is perpendicular to the tangent. Its slope is the negative reciprocal of the tangent's slope. 1. **Slope of the Normal:** \[ m_{\text{normal}} = -\frac{1}{m_{\text{tangent}}} = -\frac{1}{\frac{dy}{dx}} \] 2. **Equation of the Normal Line:** \[ y - y_0 = m_{\text{normal}} (x - x_0) \] ### **Example** **Find the tangent and normal to the polar curve \( r = 2 + 2\sin \theta \) at \( \theta = \frac{\pi}{2} \).** **Solution:** 1. **Compute \( r \) at \( \theta = \frac{\pi}{2} \):** \[ r = 2 + 2\sin\left(\frac{\pi}{2}\right) = 2 + 2(1) = 4 \] 2. **Cartesian Coordinates:** \[ x_0 = 4 \cos\left(\frac{\pi}{2}\right) = 0, \quad y_0 = 4 \sin\left(\frac{\pi}{2}\right) = 4 \] 3. **Find \( r' \):** \[ r = 2 + 2\sin \theta \implies r' = 2\cos \theta \] \[ r'\bigg|_{\theta = \frac{\pi}{2}} = 2 \cos\left(\frac{\pi}{2}\right) = 0 \] 4. **Compute \( \frac{dy}{dx} \):** \[ \frac{dy}{dx} = \frac{0 \cdot \sin\left(\frac{\pi}{2}\right) + 4 \cos\left(\frac{\pi}{2}\right)}{0 \cdot \cos\left(\frac{\pi}{2}\right) - 4 \sin\left(\frac{\pi}{2}\right)} = \frac{0 + 0}{0 - 4} = 0 \] So, the slope of the tangent is 0 (horizontal line). 5. **Equation of the Tangent:** \[ y - 4 = 0 \cdot (x - 0) \implies y = 4 \] 6. **Slope of the Normal:** \[ m_{\text{normal}} = -\frac{1}{0} \implies \text{Undefined (Vertical Line)} \] 7. **Equation of the Normal:** \[ x = 0 \] **Conclusion:** - **Tangent Line:** \( y = 4 \) - **Normal Line:** \( x = 0 \) --- ### **General Steps to Find Tangents and Normals** 1. **Express the Polar Equation \( r = f(\theta) \).** 2. **Differentiate \( r \) with respect to \( \theta \) to find \( r' = \frac{dr}{d\theta} \).** 3. **Use the formula for \( \frac{dy}{dx} \) in polar coordinates:** \[ \frac{dy}{dx} = \frac{r' \sin \theta + r \cos \theta}{r' \cos \theta - r \sin \theta} \] 4. **Calculate the slope of the tangent (\( m_{\text{tangent}} \)) and the normal (\( m_{\text{normal}} \)).** 5. **Find the Cartesian coordinates \( (x_0, y_0) \) using \( r \) and \( \theta \).** 6. **Write the equations of the tangent and normal lines using the point-slope form.** --- ## **Summary** - **De Moivre's Theorem** is essential for simplifying complex number computations and has applications in various fields of mathematics. - **Graphing Polar Equations** involves understanding the relationship between \( r \) and \( \theta \), recognizing symmetries, and plotting key points to sketch the curve accurately. - **Determining Normals and Tangents** to polar curves requires calculus techniques, specifically differentiation, and applying coordinate geometry principles to obtain the equations of these lines. Feel free to ask if you need further clarification or examples on any of these topics!