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Original
2026-01-01
Modified
2026-02-28
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<p>The complex plane is a two-dimensional plane, with the horizontal axis representing the real part and the vertical axis representing the imaginary part of the complex number. Follow these steps for the graphical representation of complex numbers in polar form. </p>
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<p>The complex plane is a two-dimensional plane, with the horizontal axis representing the real part and the vertical axis representing the imaginary part of the complex number. Follow these steps for the graphical representation of complex numbers in polar form. </p>
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<p><strong>Step 1:</strong>Find the value of the modulus The value of the modulus can be calculated using the formula, \(r = \sqrt { a^2 + b^2}\).</p>
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<p><strong>Step 1:</strong>Find the value of the modulus The value of the modulus can be calculated using the formula, \(r = \sqrt { a^2 + b^2}\).</p>
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<p><strong>Step 2:</strong>Find the value of θ The value of θ is calculated using the formula, θ = tan-1(b/a)</p>
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<p><strong>Step 2:</strong>Find the value of θ The value of θ is calculated using the formula, θ = tan-1(b/a)</p>
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<p><strong>Step 3:</strong>Find the polar form of the complex number Now we express the value of r and θ in z = r (cosθ + i sinθ)</p>
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<p><strong>Step 3:</strong>Find the polar form of the complex number Now we express the value of r and θ in z = r (cosθ + i sinθ)</p>
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<p><strong>Step 4:</strong>Graphical representation In the graph, the x-axis represents the real part and the y-axis represents the imaginary part. Mark the center (0, 0), in the direction of θ, move a distance r (modulus) from the origin at an angle θ, and mark the point. Then connect the point with the center. </p>
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<p><strong>Step 4:</strong>Graphical representation In the graph, the x-axis represents the real part and the y-axis represents the imaginary part. Mark the center (0, 0), in the direction of θ, move a distance r (modulus) from the origin at an angle θ, and mark the point. Then connect the point with the center. </p>
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<p><strong>Example:</strong></p>
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<p><strong>Example:</strong></p>
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<p>Graphically represent z = 3 + 3i</p>
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<p>Graphically represent z = 3 + 3i</p>
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<p>Find the value of modulus, \(r = \sqrt { a^2 + b^2} \)</p>
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<p>Find the value of modulus, \(r = \sqrt { a^2 + b^2} \)</p>
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<p>Here, \(r = \sqrt { 3^2 + 3^2} \)</p>
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<p>Here, \(r = \sqrt { 3^2 + 3^2} \)</p>
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<p>\(=\sqrt{ 9 +9} = {\sqrt {18}} = 3 \sqrt 2 \)</p>
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<p>\(=\sqrt{ 9 +9} = {\sqrt {18}} = 3 \sqrt 2 \)</p>
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<p>Finding the value of θ </p>
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<p>Finding the value of θ </p>
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<p>\(\theta = \tan^{-1}\left(\frac{b}{a}\right) \)</p>
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<p>\(\theta = \tan^{-1}\left(\frac{b}{a}\right) \)</p>
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<p>\(\tan^{-1}\left(\frac{3}{3}\right) = \tan^{-1}(1) = 45^\circ \)</p>
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<p>\(\tan^{-1}\left(\frac{3}{3}\right) = \tan^{-1}(1) = 45^\circ \)</p>
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<p><strong>Quadrant check:</strong>If the complex number has a negative part, the angle would be adjusted depending on the quadrant. Since both a and b are positive here, we don't need to adjust the angle. In polar form, z can be expressed as </p>
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<p><strong>Quadrant check:</strong>If the complex number has a negative part, the angle would be adjusted depending on the quadrant. Since both a and b are positive here, we don't need to adjust the angle. In polar form, z can be expressed as </p>
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<p>\(z = 3\sqrt{2} (\cos 45^\circ + i \sin 45^\circ) \)</p>
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<p>\(z = 3\sqrt{2} (\cos 45^\circ + i \sin 45^\circ) \)</p>
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<p>The argument θ of a complex number determines its direction in the complex plane. It's essential to adjust θ based on the quadrant in which the complex number lies:</p>
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<p>The argument θ of a complex number determines its direction in the complex plane. It's essential to adjust θ based on the quadrant in which the complex number lies:</p>
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<p>Quadrant I (x > 0, y > 0): \(\theta = \tan^{-1}\left(\frac{y}{x}\right) \)</p>
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<p>Quadrant I (x > 0, y > 0): \(\theta = \tan^{-1}\left(\frac{y}{x}\right) \)</p>
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<p>Quadrant II (x < 0, y > 0): \(\tan^{-1}\left(\frac{y}{x}\right) + \pi \)</p>
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<p>Quadrant II (x < 0, y > 0): \(\tan^{-1}\left(\frac{y}{x}\right) + \pi \)</p>
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<p>Quadrant III (x < 0, y < 0): \(\tan^{-1}\left(\frac{y}{x}\right) - \pi \)</p>
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<p>Quadrant III (x < 0, y < 0): \(\tan^{-1}\left(\frac{y}{x}\right) - \pi \)</p>
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<p>Quadrant IV (x > 0, y < 0): \(\tan^{-1}\left(\frac{y}{x}\right) \)</p>
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<p>Quadrant IV (x > 0, y < 0): \(\tan^{-1}\left(\frac{y}{x}\right) \)</p>
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