Advertisements
Advertisements
प्रश्न
Express the following complex in the form r(cos θ + i sin θ):
tan α − i
Advertisements
उत्तर
\[ \text { Let }z = \tan \alpha - i \]
\[ \because \tan \alpha\text { is periodic with period } \pi . \text { So, let us take } \]
\[\alpha \in [0, \frac{\pi}{2}) \cup ( \frac{\pi}{2}, \pi]\]
\[\text { Case I }: \]
\[z = \tan \alpha - i \]
\[ \Rightarrow \left| z \right| = \sqrt{\tan^2 + 1}\]
\[ = \left| \sec \alpha \right| \left[ \because 0 < \alpha < \frac{\pi}{2} \right]\]
\[ = \sec \alpha\]
\[\text { Let } \beta \text { be an acute angle given by }\tan \beta = \left| \frac{Im (z)}{Re(z)} \right|\]
\[\tan \beta = \frac{1}{\left| \tan \alpha \right|}\]
\[ = \left| \cot \alpha \right|\]
\[ = \cot \alpha\]
\[ = \tan \left( \frac{\pi}{2} - \alpha \right)\]
\[ \Rightarrow \beta = \frac{\pi}{2} - \alpha \]
\[\text { We can see that Re }(z) > 0 \text { and Im}(z) < 0 . \text { So, z lies in the fourth quadrant }. \]
\[ \therefore \arg(z) = - \beta = \alpha - \frac{\pi}{2}\]
\[\text { Thus, z in the polar form is given by }\]
\[z = \sec \alpha \left\{ \cos\left( \alpha - \frac{\pi}{2} \right) + i\sin \left( \alpha - \frac{\pi}{2} \right) \right\} \]
\[\text { Case II }: \]
\[z = \tan \alpha - i \]
\[ \Rightarrow \left| z \right| = \sqrt{\tan^2 + 1}\]
\[ = \left| \sec \alpha \right| \left[ \because \frac{\pi}{2} < \alpha < \pi \right]\]
\[ = - \sec \alpha\]
\[\text { Let } \beta \text { be an acute angle given by } \tan \beta = \left| \frac{Im (z)}{Re(z)} \right|\]
\[\tan \beta = \frac{1}{\left| \tan \alpha \right|}\]
\[ = \left| \cot \alpha \right|\]
\[ = - \cot \alpha\]
\[ = \tan \left( \alpha - \frac{\pi}{2} \right)\]
\[ \Rightarrow \beta = \alpha - \frac{\pi}{2}\]
\[\text{We can see that Re}(z) < 0 \text { and Im} (z) < 0 . So, z \text { lies in the third quadrant }. \]
\[ \therefore \arg(z) = \pi + \beta = \frac{\pi}{2} + \alpha\]
\[\text { Thus, z in the polar form is given by } \]
\[z = - \sec \alpha \left\{ \cos\left( \frac{\pi}{2} + \alpha \right) + i\sin \left( \frac{\pi}{2} + \alpha \right) \right\} \]
APPEARS IN
संबंधित प्रश्न
Express the given complex number in the form a + ib: (1 – i) – (–1 + i6)
If a + ib = `(x + i)^2/(2x^2 + 1)` prove that a2 + b2 = `(x^2 + 1)^2/(2x + 1)^2`
Evaluate the following:
\[i^{37} + \frac{1}{i^{67}}\].
Evaluate the following:
\[\left( i^{41} + \frac{1}{i^{257}} \right)^9\]
Show that 1 + i10 + i20 + i30 is a real number.
Express the following complex number in the standard form a + i b:
\[(1 + i)(1 + 2i)\]
Express the following complex number in the standard form a + i b:
\[\frac{(1 - i )^3}{1 - i^3}\]
Express the following complex number in the standard form a + i b:
\[\frac{5 + \sqrt{2}i}{1 - 2\sqrt{i}}\]
If \[z_1 = 2 - i, z_2 = - 2 + i,\] find
Re \[\left( \frac{z_1 z_2}{z_1} \right)\]
If \[z_1 = 2 - i, z_2 = - 2 + i,\] find
Im `(1/(z_1overlinez_1))`
If \[x + iy = \frac{a + ib}{a - ib}\] prove that x2 + y2 = 1.
Find the real values of θ for which the complex number \[\frac{1 + i cos\theta}{1 - 2i cos\theta}\] is purely real.
If \[\left( \frac{1 - i}{1 + i} \right)^{100} = a + ib\] find (a, b).
Evaluate the following:
\[2 x^3 + 2 x^2 - 7x + 72, \text { when } x = \frac{3 - 5i}{2}\]
Evaluate the following:
\[x^4 + 4 x^3 + 6 x^2 + 4x + 9, \text { when } x = - 1 + i\sqrt{2}\]
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):
\[\frac{1 - i}{\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}}\]
If π < θ < 2π and z = 1 + cos θ + i sin θ, then write the value of \[\left| z \right|\] .
Write the value of \[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\] .
Write the argument of −i.
If \[z = \frac{- 2}{1 + i\sqrt{3}}\],then the value of arg (z) is
The principal value of the amplitude of (1 + i) is
The least positive integer n such that \[\left( \frac{2i}{1 + i} \right)^n\] is a positive integer, is.
If \[z = \left( \frac{1 + i}{1 - i} \right)\] then z4 equals
If \[x + iy = \frac{3 + 5i}{7 - 6i},\] then y =
The value of \[(1 + i )^4 + (1 - i )^4\] is
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 the complex number \[z = x + iy\] satisfies the condition \[\left| z + 1 \right| = 1\], then z lies on
Simplify : `4sqrt(-4) + 5sqrt(-9) - 3sqrt(-16)`
Find a and b if (a + ib) (1 + i) = 2 + i
Express the following in the form of a + ib, a, b ∈ R, i = `sqrt(−1)`. State the values of a and b:
(1 + 2i)(– 2 + i)
Express the following in the form of a + ib, a, b ∈ R, i = `sqrt(−1)`. State the values of a and b:
`("i"(4 + 3"i"))/((1 - "i"))`
Express the following in the form of a + ib, a, b∈R i = `sqrt(−1)`. State the values of a and b:
`((2 + "i"))/((3 - "i")(1 + 2"i"))`
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
Find the value of `(3 + 2/i) (i^6 - i^7) (1 + i^11)`.
Evaluate the following : i35
Evaluate the following : i–888
If a = cosθ + isinθ, find the value of `(1 + "a")/(1 - "a")`.
Find the value of `(i^(592) + i^(590) + i^(588) + i^(586) + i^(584))/(i^(582) + i^(580) + i^(578) + i^(576) + i^(574))`
