A complex number \(z\) lies on the unit circle in the complex plane, as shown in the diagram.
Which of the following complex numbers is equal to \(\bar{z}\) ?
- \(-z\)
- \(z^2\)
- \(-z^3\)
- \(z^4\)
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A complex number \(z\) lies on the unit circle in the complex plane, as shown in the diagram.
Which of the following complex numbers is equal to \(\bar{z}\) ?
\(B\)
\(z=e^{-\dfrac{2i \pi}{3}}, \ \bar z=e^{\dfrac{2i \pi}{3}} \)
\(\text{By trial and error:}\)
\(z^2=e^{-\dfrac{2 \times 2i \pi}{3}} = e^{-\dfrac{4i \pi}{3}}=e^{\dfrac{2i \pi}{3}} =\bar z \)
\(\Rightarrow B\)
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i.
`text{Let}\ \z=-sqrt3+i`
`abs(z)=sqrt((-sqrt3)^2+1^2)=2`
`text(Find arg)(z):`
`tan theta=1/sqrt3\ \ =>\ \ theta=pi/6`
`=>\ text{arg}(z)=(5pi)/6`
`:.z` | `=2(cos(5pi)/6+sin(5pi)/6 i)` | |
`=2e^((5pi)/6 i)` |
ii. | `(-sqrt3+i)^(10)` | `=(2e^((5pi)/6 i))^(10)` |
`=2^10e^((50pi)/6 i)` | ||
`=1024e^(pi/3 i)` | ||
`=1024(cos(pi/3)+sin(pi/3)i)` | ||
`=1024(1/2+sqrt3/2 i)` | ||
`=512+512sqrt3 i` |
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i. | `e^(i n theta) + e^(-i n theta)` | `= cos(n theta) + i sin(n theta) + cos(-n theta) + i sin(-n theta)` |
`= cos(n theta) + i sin(n theta) + cos(n theta) – i sin(n theta)` | ||
`= 2 cos (n theta)` |
ii. | `(e^{i theta} + e^{-i theta})^4` | `= (2 cos theta)^4` |
`= 16 cos^4 theta` |
`text{Expand} \ (e^{i theta} + e^{-i theta})^4 :`
`e^(i 4 theta) + 4 e^(i 3 theta) e^(-i theta) + 6 e^(i 2 theta) e^(-i 2 theta) + 4 e^(i theta) e^(-i 3 theta) + e^(-i 4 theta)`
`= e^(i 4 theta) + 4e^(i 2 theta) + 6 + 4^(-i 2 theta) + e^(-i 4 theta)`
`= e^(i 4 theta) + e^(i 4 theta) + 4 (e^{i 2 theta} + e^{-i 2 theta}) + 6`
`= 2 cos (4 theta) + 8 cos (2 theta) + 6`
`therefore \ 16 cos^4 theta` | `= 2 cos (4 theta) + 8 cos (2 theta) + 6` |
`cos^4 theta` | `= frac{1}{8} cos(4 theta) + 1/2 cos(2 theta) + 3/8` |
`cos^4 theta` | `= frac{1}{8} (cos(4 theta) + 4 cos(2 theta) + 3)` |
iii. | `int_0^(frac{pi}{2}) cos^4 theta\ d theta` | `= frac{1}{8} int_0^(frac{pi}{2}) cos(4 theta) + 4 cos(2 theta) + 3\ d theta` |
`= frac{1}{8} [ frac{1}{4} sin(4 theta) + 2 sin (2 theta) + 3 theta ]_0^(frac{pi}{2}` | ||
`= frac{1}{8} [( frac{1}{4} sin (2 pi) + 2 sin pi + frac{3 pi}{2}) – 0 ]` | ||
`= frac{1}{8} ( frac{3 pi}{2})` | ||
`= frac{3 pi}{16}` |