Advertisements
Advertisements
Question
\[\text { If }z = 1 - \text { cos }\theta + i \text { sin }\theta, \text { then } \left| z \right| =\]
Options
\[2 \sin\frac{\theta}{2}\]
\[2 \cos\frac{\theta}{2}\]
\[2\left| \sin\frac{\theta}{2} \right|\]
\[2\left| \cos\frac{\theta}{2} \right|\]
Solution
\[2\left| \sin\frac{\theta}{2} \right|\]
\[\because z = 1 - \cos\theta + i \sin\theta\]
\[ \Rightarrow \left| z \right| = \sqrt{\left( 1 - \cos\theta \right)^2 + \sin^2 \theta}\]
\[ \Rightarrow \left| z \right| = \sqrt{1 + \cos^2 \theta - 2\cos\theta + \sin^2 \theta}\]
\[ \Rightarrow \left| z \right| = \sqrt{1 + 1 - 2\cos\theta}\]
\[ \Rightarrow \left| z \right| = \sqrt{2\left( 1 - \cos\theta \right)}\]
\[ \Rightarrow \left| z \right| = \sqrt{4 \sin^2 \frac{\theta}{2}}\]
\[ \Rightarrow \left| z \right|=2\left| \sin\frac{\theta}{2} \right|\]
APPEARS IN
RELATED QUESTIONS
Express the given complex number in the form a + ib: (1 – i) – (–1 + i6)
Express the given complex number in the form a + ib: (1 – i)4
Evaluate the following:
\[i^{37} + \frac{1}{i^{67}}\].
Evaluate the following:
\[i^{30} + i^{40} + i^{60}\]
Find the value of the following expression:
i + i2 + i3 + i4
Find the value of the following expression:
1+ i2 + i4 + i6 + i8 + ... + i20
Express the following complex number in the standard form a + i b:
\[\frac{3 + 2i}{- 2 + i}\]
Express the following complex number in the standard form a + i b:
\[\frac{2 + 3i}{4 + 5i}\]
Evaluate the following:
\[2 x^3 + 2 x^2 - 7x + 72, \text { when } x = \frac{3 - 5i}{2}\]
Evaluate the following:
\[2 x^4 + 5 x^3 + 7 x^2 - x + 41, \text { when } x = - 2 - \sqrt{3}i\]
For a positive integer n, find the value of \[(1 - i )^n \left( 1 - \frac{1}{i} \right)^n\].
If \[\left( 1 + i \right)z = \left( 1 - i \right) \bar{z}\],then show that \[z = - i \bar{z}\].
Solve the system of equations \[\text { Re }\left( z^2 \right) = 0, \left| z \right| = 2\].
If \[\left| z + 1 \right| = z + 2\left( 1 + i \right)\],find z.
Find the number of solutions of \[z^2 + \left| z \right|^2 = 0\].
Write (i25)3 in polar form.
Express the following complex in the form r(cos θ + i sin θ):
tan α − i
Express \[\sin\frac{\pi}{5} + i\left( 1 - \cos\frac{\pi}{5} \right)\] in polar form.
Write −1 + i \[\sqrt{3}\] in polar form .
If \[\left| z + 4 \right| \leq 3\], then find the greatest and least values of \[\left| z + 1 \right|\].
If \[\left| z \right| = 2 \text { and } \arg\left( z \right) = \frac{\pi}{4}\],find z.
If\[z = \cos\frac{\pi}{4} + i \sin\frac{\pi}{6}\], then
If \[z = \frac{- 2}{1 + i\sqrt{3}}\],then the value of arg (z) is
If \[z = \frac{1}{1 - cos\theta - i sin\theta}\] then Re (z) =
A real value of x satisfies the equation \[\frac{3 - 4ix}{3 + 4ix} = a - ib (a, b \in \mathbb{R}), if a^2 + b^2 =\]
If z is a complex number, then
Which of the following is correct for any two complex numbers z1 and z2?
Express the following in the form of a + ib, a, b∈R i = `sqrt(−1)`. State the values of a and b:
`(3 + 2"i")/(2 - 5"i") + (3 -2"i")/(2 + 5"i")`
Express the following in the form of a + ib, a, b∈R i = `sqrt(−1)`. State the values of a and b:
`(2 + sqrt(-3))/(4 + sqrt(-3))`
Express the following in the form of a + ib, a, b ∈ R i = `sqrt(−1)`. State the values of a and b:
`(- sqrt(5) + 2sqrt(-4)) + (1 -sqrt(-9)) + (2 + 3"i")(2 - 3"i")`
Show that `(-1 + sqrt(3)"i")^3` is a real number
Evaluate the following : i116
Evaluate the following : i403
Evaluate the following : `1/"i"^58`
Evaluate the following : i30 + i40 + i50 + i60
If a = cosθ + isinθ, find the value of `(1 + "a")/(1 - "a")`.
Show that `(-1 + sqrt3 "i")^3` is a real number.
