Advertisements
Advertisements
प्रश्न
Express the following complex in the form r(cos θ + i sin θ):
1 + i tan α
उत्तर
\[\text{Let } z = 1 + i\tan \alpha \]
\[ \because \tan \alpha\text { is periodic with period }π. \text { So, let us take } \]
\[\alpha \in [0,\frac{\pi}{2}) \cup ( \frac{\pi}{2}, \pi]\]
\[Case I: \]
\[\text { When } \alpha \in [0, \frac{\pi}{2})\]
\[z = 1 + i\tan \alpha \]
\[ \Rightarrow \left| z \right| = \sqrt{1 + \tan^2 \alpha}\]
\[ = \left| \sec \alpha \right| \left[ \because 0 < \alpha < \frac{\pi}{2} \right]\]
\[ = \sec \alpha\]
\[\text { Let } \beta \text { be an acute angle given by } \tan \beta = \left| \frac{Im (z)}{Re(z)} \right|\]
\[\tan \beta = \left| \tan \alpha \right|\]
\[ = \tan \alpha\]
\[ \Rightarrow \beta = \alpha \]
\[\text { As z lies in the first quadrant . Therefore}, \arg(z) = \beta = \alpha\]
\[\text { Thus, z in the polar form is given by } \]
\[z = \sec \alpha \left( \cos\alpha + i\sin \alpha \right)\]
\[\text{Case II }: \]
\[z = 1 + i \tan \alpha \]
\[ \Rightarrow \left| z \right| = \sqrt{1 + \tan^2 \alpha}\]
\[ = \left| \sec \alpha \right| \left[ \because \frac{\pi}{2} < \alpha < \pi \right]\]
\[ = - \sec \alpha\]
\[\text { Let } \beta \text { be an acute angle given by } \tan \beta = \left| \frac{Im (z)}{Re(z)} \right|\]
\[\tan \beta = \left| \tan \alpha \right|\]
\[ = - \tan \alpha\]
\[ \Rightarrow \tan \beta = \tan \left( \pi - \alpha \right)\]
\[ \Rightarrow \beta = \pi - \alpha\]
\[\text { As, z lies in the fourth quadrant } . \]
\[ \therefore \arg(z) = - \beta = \alpha - \pi\]
\[\text { Thus, z in the polar form is given by } \]
\[z = - \sec \alpha \left\{ \cos\left( \alpha - \pi \right) + i\sin \left( \alpha - \pi \right) \right\} \]
APPEARS IN
संबंधित प्रश्न
Express the given complex number in the form a + ib: `(5i) (- 3/5 i)`
Express the given complex number in the form a + ib: i–39
Express the given complex number in the form a + ib: `(1/3 + 3i)^3`
Express the given complex number in the form a + ib: `(-2 - 1/3 i)^3`
Evaluate the following:
\[( i^{77} + i^{70} + i^{87} + i^{414} )^3\]
Find the value of the following expression:
\[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\]
Find the value of the following expression:
(1 + i)6 + (1 − i)3
Express the following complex number in the standard form a + i b:
\[\frac{(1 + i)(1 + \sqrt{3}i)}{1 - i}\] .
Express the following complex number in the standard form a + i b:
\[\frac{2 + 3i}{4 + 5i}\]
Express the following complex number in the standard form a + i b:
\[\frac{3 - 4i}{(4 - 2i)(1 + i)}\]
Express the following complex number in the standard form a + i b:
\[\frac{5 + \sqrt{2}i}{1 - 2\sqrt{i}}\]
Find the multiplicative inverse of the following complex number:
1 − i
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 \[a = \cos\theta + i\sin\theta\], find the value of \[\frac{1 + a}{1 - a}\].
Express the following complex in the form r(cos θ + i sin θ):
tan α − i
If π < θ < 2π and z = 1 + cos θ + i sin θ, then write the value of \[\left| z \right|\] .
Write the argument of −i.
Find the real value of a for which \[3 i^3 - 2a i^2 + (1 - a)i + 5\] is real.
If `(3+2i sintheta)/(1-2 i sin theta)`is a real number and 0 < θ < 2π, then θ =
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 \[z = \frac{1 + 2i}{1 - (1 - i )^2}\], then arg (z) equal
If \[z = \frac{1 + 7i}{(2 - i )^2}\] , then
\[\frac{1 + 2i + 3 i^2}{1 - 2i + 3 i^2}\] equals
If z is a complex number, then
Find a and b if a + 2b + 2ai = 4 + 6i
Express the following in the form of a + ib, a, b∈R i = `sqrt(−1)`. State the values of a and b:
`((1 + "i")/(1 - "i"))^2`
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))`
Show that `(-1 + sqrt(3)"i")^3` is a real number
Evaluate the following : i888
Evaluate the following : `1/"i"^58`
Evaluate the following : i–888
Show that 1 + i10 + i20 + i30 is a real number
Answer the following:
Show that z = `5/((1 - "i")(2 - "i")(3 - "i"))` is purely imaginary number.
If z1 and z2 both satisfy `z + barz = 2|z - 1|` arg`(z_1 - z_2) = pi/4`, then find `"Im" (z_1 + z_2)`.
If a = cosθ + isinθ, find the value of `(1 + "a")/(1 - "a")`.
The real value of θ for which the expression `(1 + i cos theta)/(1 - 2i cos theta)` is a real number is ______.
Find the value of `(i^(592) + i^(590) + i^(588) + i^(586) + i^(584))/(i^(582) + i^(580) + i^(578) + i^(576) + i^(574))`
