1. Discuss De Moivres theorem of comple numbers 2. Discuss graphs of given polar equations and normal and tangents are determined using co-ordinate goometry.

Certainly! Let's delve into each of your queries systematically: --- ## 1. **De Moivre's Theorem of Complex Numbers** **Overview:** De Moivre's Theorem is a fundamental result in complex number theory, linking complex numbers and trigonometry. It provides a powerful method to compute powers and roots of complex numbers expressed in polar (trigonometric) form. **Statement of De Moivre's Theorem:** For any real number \( \theta \) and any integer \( n \), De Moivre's Theorem states that: \[ (\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta) \] where \( i \) is the imaginary unit, \( i^2 = -1 \). **Proof Sketch:** The theorem can be proven using mathematical induction and the binomial theorem. 1. **Base Case (n=1):** \[ (\cos \theta + i \sin \theta)^1 = \cos \theta + i \sin \theta \] Which is trivially equal to the right side of the theorem for \( n=1 \). 2. **Inductive Step:** Assume the theorem holds for \( n \), i.e., \[ (\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta) \] Then for \( n+1 \): \[ (\cos \theta + i \sin \theta)^{n+1} = (\cos \theta + i \sin \theta)^n (\cos \theta + i \sin \theta) \] Using the induction hypothesis and applying trigonometric identities (specifically, the angle addition formulas), we can show: \[ \cos((n+1)\theta) + i \sin((n+1)\theta) \] Thus, the theorem holds for \( n+1 \), completing the induction. **Applications:** 1. **Computing Powers of Complex Numbers:** It simplifies raising complex numbers to integer powers, especially when expressed in polar form. *Example:* Compute \( (\cos 30^\circ + i \sin 30^\circ)^3 \): \[ \cos(3 \times 30^\circ) + i \sin(3 \times 30^\circ) = \cos 90^\circ + i \sin 90^\circ = 0 + i \times 1 = i \] 2. **Finding Roots of Complex Numbers:** De Moivre's theorem can be extended to find the \( n \)-th roots of a complex number. *Example:* To find the cube roots of 1: \[ 1 = \cos 0 + i \sin 0 \] The cube roots are: \[ \cos\left(\frac{0 + 2k\pi}{3}\right) + i \sin\left(\frac{0 + 2k\pi}{3}\right), \quad k = 0, 1, 2 \] Which yields the three cube roots of unity. 3. **Relation to Euler's Formula:** When expressed using Euler's formula, \( e^{i\theta} = \cos \theta + i \sin \theta \), De Moivre's theorem becomes: \[ (e^{i\theta})^n = e^{in\theta} \] This forms a bridge between trigonometry and exponential functions in complex analysis. --- ## 2. **Graphs of Given Polar Equations and Determining Normals and Tangents Using Coordinate Geometry** **Overview:** Polar coordinates represent points in the plane using a radius and an angle, offering an alternative to Cartesian coordinates. Graphing polar equations and finding their normals and tangents involve translating between polar and Cartesian forms and applying techniques from coordinate geometry. ### **A. Graphing Polar Equations** Polar equations can describe a variety of curves, some of which are more naturally expressed in polar form, such as circles, spirals, and roses. **Common Polar Curves:** 1. **Circles:** - \( r = a \) (a circle of radius \( a \) centered at the origin) - \( r = 2a \cos \theta \) or \( r = 2a \sin \theta \) (circles passing through the origin) 2. **Roses:** - \( r = a \cos(k\theta) \) or \( r = a \sin(k\theta) \) - Number of petals depends on \( k \): - If \( k \) is odd, \( k \) petals. - If \( k \) is even, \( 2k \) petals. 3. **Spirals:** - **Archimedean Spiral:** \( r = a + b\theta \) - **Logarithmic Spiral:** \( r = a e^{b\theta} \) 4. **Lemniscates:** - \( r^2 = a^2 \cos(2\theta) \) (a figure-eight shape) **Plotting Steps:** 1. **Choose Values for \( \theta \):** Select a range of angles \( \theta \), typically from \( 0 \) to \( 2\pi \). 2. **Compute Corresponding \( r \):** For each \( \theta \), calculate the radius \( r \). 3. **Plot Points:** Plot the points \( (r, \theta) \) on polar graph paper or using software. 4. **Connect Points Smoothly:** Sketch the curve by connecting the plotted points, observing symmetry where applicable. **Example:** Plot \( r = 2 \cos \theta \). - At \( \theta = 0 \), \( r = 2 \). - At \( \theta = \frac{\pi}{2} \), \( r = 0 \). - At \( \theta = \pi \), \( r = -2 \) (which is equivalent to \( r = 2 \) and \( \theta = 0 \) due to the negative radius). This yields a circle of radius 1 centered at \( (1, 0) \) in Cartesian coordinates. ### **B. Determining Normals and Tangents Using Coordinate Geometry** To find normals and tangents to a polar curve, it's often helpful to convert the polar equation to Cartesian coordinates. However, calculus-based methods are typically used for finding derivatives and hence tangents and normals. Since the query specifies using coordinate geometry, we'll outline a strategy that minimizes reliance on calculus. **Steps to Find Tangent and Normal Lines:** 1. **Convert Polar to Cartesian Coordinates:** Use the relationships: \[ x = r \cos \theta, \quad y = r \sin \theta \] Given a polar equation \( r = f(\theta) \), express \( x \) and \( y \) in terms of \( \theta \). 2. **Parametric Representation:** Represent the curve parametrically: \[ x(\theta) = f(\theta) \cos \theta, \quad y(\theta) = f(\theta) \sin \theta \] 3. **Compute Derivatives:** While purely coordinate geometry avoids calculus, finding slopes typically requires differentiation. However, assuming basic differentiation is allowed: - First derivatives: \[ \frac{dx}{d\theta} = f'(\theta) \cos \theta - f(\theta) \sin \theta \] \[ \frac{dy}{d\theta} = f'(\theta) \sin \theta + f(\theta) \cos \theta \] - Slope of the tangent line \( m_t \): \[ m_t = \frac{dy/d\theta}{dx/d\theta} \] - Slope of the normal line \( m_n \): \[ m_n = -\frac{1}{m_t} \] 4. **Point of Tangency:** Choose a specific \( \theta \) value where you want to find the tangent and normal lines. Compute \( x \) and \( y \) at this \( \theta \) to find the point of tangency. 5. **Form Equations of Tangent and Normal Lines:** Using the point-slope form: - **Tangent Line:** \[ y - y_0 = m_t (x - x_0) \] - **Normal Line:** \[ y - y_0 = m_n (x - x_0) \] where \( (x_0, y_0) \) is the point of tangency. **Example:** Find the tangent and normal lines to the polar curve \( r = a(1 + \cos \theta) \) at \( \theta = 0 \). 1. **Convert to Cartesian:** \[ x = a(1 + \cos \theta) \cos \theta = a(1 + \cos \theta)\cos \theta \] \[ y = a(1 + \cos \theta) \sin \theta \] At \( \theta = 0 \): \[ x = a(1 + 1) \times 1 = 2a, \quad y = 0 \] Point of tangency: \( (2a, 0) \) 2. **Compute Derivatives:** \[ f(\theta) = a(1 + \cos \theta), \quad f'(\theta) = -a \sin \theta \] \[ \frac{dx}{d\theta} = -a \sin \theta \cos \theta - a(1 + \cos \theta) \sin \theta = -a \sin \theta (2 \cos \theta + 1) \] \[ \frac{dy}{d\theta} = -a \sin \theta \sin \theta + a(1 + \cos \theta) \cos \theta = a \cos \theta (1 + \cos \theta) - a \sin^2 \theta \] At \( \theta = 0 \): \[ \frac{dx}{d\theta} = 0, \quad \frac{dy}{d\theta} = a(1 + 1) - 0 = 2a \] Thus, the slope of the tangent: \[ m_t = \frac{dy/d\theta}{dx/d\theta} = \frac{2a}{0} \rightarrow \text{Vertical Tangent} \] Therefore, the tangent line is vertical: \( x = 2a \) 3. **Normal Line:** Since the tangent is vertical, the normal is horizontal. \[ y = 0 \] **Visualization:** For the given curve \( r = a(1 + \cos \theta) \), known as the cardioid, at \( \theta = 0 \), the tangent line is the vertical line \( x = 2a \), and the normal line is the horizontal line \( y = 0 \). **Alternative Approach Without Calculus:** For certain symmetrical curves or specific points, you might determine tangents and normals geometrically by analyzing symmetry, leveraging known properties, or using geometric constructions. However, for general polar curves and arbitrary points, calculus provides a systematic method. --- **Conclusion:** Understanding De Moivre's Theorem facilitates complex number manipulations, especially powers and roots in polar form, while mastering the graphing and analysis of polar equations enhances the ability to work with a diverse range of curves and their geometric properties. Combining both algebraic and geometric techniques allows for comprehensive exploration and problem-solving in complex and polar coordinate systems.