Express the following complex numbers in the form r cos- - isin- -1 - sin - - i cos -

r=√((1 sinα)^2+cos ^2α)=√(2 2sinα)=√(2)√(1 sinα)=√(2)( sinα2+cosα2) 1 sinα+icosα=(√(1 sinα))^2+i(cos^2α2 sin^2α2)=√(2)(√(1 sinα))(√(1 sinα)√(2)+i√(1+s ...

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Standard XII

Mathematics

Polar Representation of a Complex Number

Express the f...

Question

Express the following complex numbers in the form $r (cos θ + i sin θ)$ :
1 - sin $α$ + i cos $α$

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Solution

$r = \sqrt{(1 - sin α)^{2} + {cos}^{2} α} = \sqrt{2 - 2 sin α} = \sqrt{2} \sqrt{1 - sin α} = \sqrt{2} (- sin \frac{α}{2} + cos \frac{α}{2})$
$1 - sin α + i cos α = (\sqrt{1 - sin α})^{2} + i ({cos}^{2} \frac{α}{2} - {sin}^{2} \frac{α}{2}) = \sqrt{2} (\sqrt{1 - sin α}) (\frac{\sqrt{1 - sin α}}{\sqrt{2}} + i \frac{\sqrt{1 + sin α}}{\sqrt{2}})$

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Express the following complex numbers in the form r(cosθ+isinθ) :1 - sinα + i cosα

r=√(1−sinα)2+cos2α=√2−2sinα=√2√1−sinα=√2(−sinα2+cosα2)1−sinα+icosα=(√1−sinα)2+i(cos2α2−sin2α2)=√2(√1−sinα)(√1−sinα√2+i√1+sinα√2)

Express the following complex numbers in the form $r (cos θ + i sin θ)$ :
1 - sin $α$ + i cos $α$