Advertisements
Advertisements
प्रश्न
Write the argument of \[\left( 1 + i\sqrt{3} \right)\left( 1 + i \right)\left( \cos\theta + i\sin\theta \right)\].
Disclaimer: There is a misprinting in the question. It should be \[\left( 1 + i\sqrt{3} \right)\] instead of \[\left( 1 + \sqrt{3} \right)\].
Advertisements
उत्तर
Let the argument of \[\left( 1 + i\sqrt{3} \right)\] be α. Then,
\[\tan\alpha = \frac{\sqrt{3}}{1} = \tan\frac{\pi}{3}\]
\[ \Rightarrow \alpha = \frac{\pi}{3}\]
Let the argument of \[\left( 1 + i \right)\] be β. Then,
\[\text { tan }\beta = \frac{1}{1} = \tan\frac{\pi}{4}\]
\[ \Rightarrow \beta = \frac{\pi}{4}\]
Let the argument of \[\left( cos\theta + isin\theta \right)\] be γ. Then,
\[\text { tan }\gamma = \frac{sin\theta}{cos\theta} = \text { tan }\theta\]
\[ \Rightarrow \gamma = \theta\]
∴ The argument of
\[\left( 1 + i\sqrt{3} \right)\left( 1 + i \right)\left( cos\theta + isin\theta \right) = \alpha + \beta + \gamma = \frac{\pi}{3} + \frac{\pi}{4} + \theta = \frac{7\pi}{12} + \theta\]
Hence, the argument of
\[\left( 1 + i\sqrt{3} \right)\left( 1 + i \right)\left( cos\theta + isin\theta \right) is \frac{7\pi}{12} + \theta\]
APPEARS IN
संबंधित प्रश्न
Express the given complex number in the form a + ib: 3(7 + i7) + i(7 + i7)
Express the given complex number in the form a + ib: `(1/3 + 3i)^3`
Let z1 = 2 – i, z2 = –2 + i. Find `"Im"(1/(z_1barz_1))`
Evaluate the following:
\[\frac{1}{i^{58}}\]
Express the following complex number in the standard form a + i b:
\[\frac{3 - 4i}{(4 - 2i)(1 + i)}\]
If \[x + iy = \frac{a + ib}{a - ib}\] prove that x2 + y2 = 1.
Find the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
Find the real values of θ for which the complex number \[\frac{1 + i cos\theta}{1 - 2i cos\theta}\] is purely real.
If \[\frac{\left( 1 + i \right)^2}{2 - i} = x + iy\] find x + y.
If \[\left( \frac{1 - i}{1 + i} \right)^{100} = a + ib\] find (a, b).
If \[\left( 1 + i \right)z = \left( 1 - i \right) \bar{z}\],then show that \[z = - i \bar{z}\].
If z1 is a complex number other than −1 such that \[\left| z_1 \right| = 1\] and \[z_2 = \frac{z_1 - 1}{z_1 + 1}\] then show that the real parts of z2 is zero.
Express the following complex in the form r(cos θ + i sin θ):
1 − sin α + i cos α
If z1 and z2 are two complex numbers such that \[\left| z_1 \right| = \left| z_2 \right|\] and arg(z1) + arg(z2) = \[\pi\] then show that \[z_1 = - \bar{{z_2}}\].
Write −1 + i \[\sqrt{3}\] in polar form .
Write the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
If \[\frac{\left( a^2 + 1 \right)^2}{2a - i} = x + iy\] find the value of \[x^2 + y^2\].
Find the real value of a for which \[3 i^3 - 2a i^2 + (1 - a)i + 5\] is real.
If \[z = \frac{- 2}{1 + i\sqrt{3}}\],then the value of arg (z) is
The least positive integer n such that \[\left( \frac{2i}{1 + i} \right)^n\] is a positive integer, is.
\[(\sqrt{- 2})(\sqrt{- 3})\] is equal to
The argument of \[\frac{1 - i\sqrt{3}}{1 + i\sqrt{3}}\] is
If \[x + iy = \frac{3 + 5i}{7 - 6i},\] then y =
The amplitude of \[\frac{1}{i}\] is equal to
\[\frac{1 + 2i + 3 i^2}{1 - 2i + 3 i^2}\] equals
If \[f\left( z \right) = \frac{7 - z}{1 - z^2}\] , where \[z = 1 + 2i\] then \[\left| f\left( z \right) \right|\] is
Find a and b if (a – b) + (a + b)i = a + 5i
Find a and b if abi = 3a − b + 12i
Express the following in the form of a + ib, a, b∈R i = `sqrt(−1)`. State the values of a and b:
(1 + i)−3
Evaluate the following : i403
If `((1 + "i"sqrt3)/(1 - "i"sqrt3))^"n"` is an integer, then n is ______.
If z1 = 3 – 2i and z2 = –1 + 3i, then Im(z1z2) = ______.
If w is a complex cube-root of unity, then prove the following
(w2 + w − 1)3 = −8
