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Q1. Use trigonometric identities to prove each of the following below. i. cos x / (1 + cos x) + (1+sin x)/cos x = 2Sec x ii. sin x + tan x / (1 + cos x) = Tan x. Q2. Complete the following table. Q3. Expand (2x-y)6

Mathematics

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Step-by-step solution

13 steps

Here are the solutions for completing the table of complex numbers:

Q2. Complete the following table.

x+yix+yimodulus (rr)argument (θ\theta)polar formexp form
(a)−2+2i3-2 + 2i\sqrt{3}42π3\frac{2\pi}{3}4(cos⁡2π3+isin⁡2π3)4\left(\cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3}\right)4ei2π34e^{i\frac{2\pi}{3}}
(b)−322−i322-\frac{3\sqrt{2}}{2} - i\frac{3\sqrt{2}}{2}3−3π4-\frac{3\pi}{4}3(cos⁡(−3π4)+isin⁡(−3π4))3\left(\cos \left(-\frac{3\pi}{4}\right) + i \sin \left(-\frac{3\pi}{4}\right)\right)3e−i3π43e^{-i\frac{3\pi}{4}}
(c)33+3i3\sqrt{3} + 3i6π6\frac{\pi}{6}6(cos⁡π6+isin⁡π6)6\left(\cos \frac{\pi}{6} + i \sin \frac{\pi}{6}\right)6eiπ66e^{i\frac{\pi}{6}}
(d)πi\pi iπ\piπ2\frac{\pi}{2}π(cos⁡π2+isin⁡π2)\pi\left(\cos \frac{\pi}{2} + i \sin \frac{\pi}{2}\right)πeiπ2\pi e^{i\frac{\pi}{2}}

Here are the step-by-step calculations for each row:

Part a· 4 steps

Given x+yi=−2+2i3x+yi = -2 + 2i\sqrt{3}

  1. Calculate the modulus rr.

    r=x2+y2=(−2)2+(23)2=4+12=16=4r = \sqrt{x^2 + y^2} = \sqrt{(-2)^2 + (2\sqrt{3})^2} = \sqrt{4 + 12} = \sqrt{16} = \boxed{4}

  2. Calculate the argument θ\theta.

    Since x<0x < 0 and y>0y > 0, the complex number is in the second quadrant.

    tan⁡α=∣yx∣=∣23−2∣=3⇒α=π3\tan \alpha = \left|\frac{y}{x}\right| = \left|\frac{2\sqrt{3}}{-2}\right| = \sqrt{3} \Rightarrow \alpha = \frac{\pi}{3} θ=π−α=π−π3=2π3\theta = \pi - \alpha = \pi - \frac{\pi}{3} = \boxed{\frac{2\pi}{3}}

  3. Write the polar form.

    r(cos⁡θ+isin⁡θ)=4(cos⁡2π3+isin⁡2π3)r(\cos \theta + i \sin \theta) = \boxed{4\left(\cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3}\right)}

  4. Write the exponential form.

    reiθ=4ei2π3re^{i\theta} = \boxed{4e^{i\frac{2\pi}{3}}}

Part b· 3 steps

Given modulus r=3r = 3 and argument θ=−3π4\theta = -\frac{3\pi}{4}

  1. Write the polar form.

    r(cos⁡θ+isin⁡θ)=3(cos⁡(−3π4)+isin⁡(−3π4))r(\cos \theta + i \sin \theta) = \boxed{3\left(\cos \left(-\frac{3\pi}{4}\right) + i \sin \left(-\frac{3\pi}{4}\right)\right)}

  2. Calculate x+yix+yi.

    x=rcos⁡θ=3cos⁡(−3π4)=3(−22)=−322x = r \cos \theta = 3 \cos\left(-\frac{3\pi}{4}\right) = 3 \left(-\frac{\sqrt{2}}{2}\right) = -\frac{3\sqrt{2}}{2} y=rsin⁡θ=3sin⁡(−3π4)=3(−22)=−322y = r \sin \theta = 3 \sin\left(-\frac{3\pi}{4}\right) = 3 \left(-\frac{\sqrt{2}}{2}\right) = -\frac{3\sqrt{2}}{2} x+yi=−322−i322x+yi = \boxed{-\frac{3\sqrt{2}}{2} - i\frac{3\sqrt{2}}{2}}

  3. Write the exponential form.

    reiθ=3e−i3π4re^{i\theta} = \boxed{3e^{-i\frac{3\pi}{4}}}

Part c· 3 steps

Given polar form 6(cos⁡π6+isin⁡π6)6\left(\cos \frac{\pi}{6} + i \sin \frac{\pi}{6}\right)

  1. Identify the modulus and argument from the polar form.

    r=6r = \boxed{6} θ=π6\theta = \boxed{\frac{\pi}{6}}

  2. Calculate x+yix+yi.

    x=rcos⁡θ=6cos⁡(π6)=6(32)=33x = r \cos \theta = 6 \cos\left(\frac{\pi}{6}\right) = 6 \left(\frac{\sqrt{3}}{2}\right) = 3\sqrt{3} y=rsin⁡θ=6sin⁡(π6)=6(12)=3y = r \sin \theta = 6 \sin\left(\frac{\pi}{6}\right) = 6 \left(\frac{1}{2}\right) = 3 x+yi=33+3ix+yi = \boxed{3\sqrt{3} + 3i}

  3. Write the exponential form.

    reiθ=6eiπ6re^{i\theta} = \boxed{6e^{i\frac{\pi}{6}}}

Part d· 3 steps

Given exponential form πeiπ2\pi e^{i\frac{\pi}{2}}

  1. Identify the modulus and argument from the exponential form.

    r=πr = \boxed{\pi} θ=π2\theta = \boxed{\frac{\pi}{2}}

  2. Write the polar form.

    r(cos⁡θ+isin⁡θ)=π(cos⁡π2+isin⁡π2)r(\cos \theta + i \sin \theta) = \boxed{\pi\left(\cos \frac{\pi}{2} + i \sin \frac{\pi}{2}\right)}

  3. Calculate x+yix+yi.

    x=rcos⁡θ=πcos⁡(π2)=π⋅0=0x = r \cos \theta = \pi \cos\left(\frac{\pi}{2}\right) = \pi \cdot 0 = 0 y=rsin⁡θ=πsin⁡(π2)=π⋅1=πy = r \sin \theta = \pi \sin\left(\frac{\pi}{2}\right) = \pi \cdot 1 = \pi x+yi=πix+yi = \boxed{\pi i}
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