If z-1-cos-i sin- then -z-A- 2 sin-2B- 2-sin-2-C- -cos-2-D- 2 cos-2

The correct option is B 2|sin θ/2| ∵ z=1 cos θ+i sin θ ⇒ |z|=√((1 cos θ)^2+sin^2θ) ⇒ |z|=√(1+cos^2θ 2cos θ+sin^2θ) ⇒ |z|=√(1+1 2cos θ) ⇒ |z|=√(2(1 cos θ)) ⇒ |z ...

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Byju's Answer

Standard XII

Mathematics

Modulus of a Complex Number

If z=1-cosθ+i...

Question

$If z = 1 - c o s θ + i s i n θ, then | z | =$

$2 s i n \frac{θ}{2}$

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$2 c o s \frac{θ}{2}$

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$2 ∣ ∣ s i n \frac{θ}{2} ∣ ∣$

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$∣ ∣ c o s \frac{θ}{2} ∣ ∣$

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Solution

The correct option is C
$2 ∣ ∣ s i n \frac{θ}{2} ∣ ∣$

$∵ z = 1 - c o s θ + i s i n θ \Rightarrow | z | = \sqrt{(1 - c o s θ)^{2} + s i n^{2} θ} \Rightarrow | z | = \sqrt{1 + c o s^{2} θ - 2 c o s θ + s i n^{2} θ} \Rightarrow | z | = \sqrt{1 + 1 - 2 c o s θ} \Rightarrow | z | = \sqrt{2 (1 - c o s θ)} \Rightarrow | z | = \sqrt{4 s i n^{2} \frac{θ}{2}} \Rightarrow | z | = 2 ∣ ∣ s i n \frac{θ}{2} ∣ ∣$

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Similar questions

Q. Assertion :If

tan (\frac{π}{2} sin θ) = cot (\frac{π}{2} cos θ)

, then

sin θ + cos θ = \sqrt{2}

Reason:

- \sqrt{2} \leq sin θ + cos θ \leq \sqrt{2}

Q. If

sin θ + 2 cos θ = 1

, then prove that

2 sin θ - cos θ = 2

Q. Question 5
If

s i n θ + 2 c o s θ = 1

, then prove that

2 s i n θ - c o s θ = 2

Q. If

sin θ + 2 cos θ = 1

Whether the given equation

2 sin θ - cos θ = 2

is ?

Q. If cos θ + sin θ =

\sqrt{2}

cos θ, show that cos θ − sin θ =

\sqrt{2}

sin θ.

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If z=1−cos θ+i sin θ, then |z|=

The correct option is C 2∣∣sin θ2∣∣ ∵ z=1−cos θ+i sin θ⇒ |z|=√(1−cos θ)2+sin2θ⇒ |z|=√1+cos2θ−2cos θ+sin2θ⇒ |z|=√1+1−2cos θ⇒ |z|=√2(1−cos θ)⇒ |z|=√4sin2θ2 ⇒ |z|=2∣∣sin θ2∣∣

$If z = 1 - c o s θ + i s i n θ, then | z | =$