Advertisements
Advertisements
Question
Express the following complex in the form r(cos θ + i sin θ):
tan α − i
Advertisements
Solution
\[ \text { Let }z = \tan \alpha - i \]
\[ \because \tan \alpha\text { is periodic with period } \pi . \text { So, let us take } \]
\[\alpha \in [0, \frac{\pi}{2}) \cup ( \frac{\pi}{2}, \pi]\]
\[\text { Case I }: \]
\[z = \tan \alpha - i \]
\[ \Rightarrow \left| z \right| = \sqrt{\tan^2 + 1}\]
\[ = \left| \sec \alpha \right| \left[ \because 0 < \alpha < \frac{\pi}{2} \right]\]
\[ = \sec \alpha\]
\[\text { Let } \beta \text { be an acute angle given by }\tan \beta = \left| \frac{Im (z)}{Re(z)} \right|\]
\[\tan \beta = \frac{1}{\left| \tan \alpha \right|}\]
\[ = \left| \cot \alpha \right|\]
\[ = \cot \alpha\]
\[ = \tan \left( \frac{\pi}{2} - \alpha \right)\]
\[ \Rightarrow \beta = \frac{\pi}{2} - \alpha \]
\[\text { We can see that Re }(z) > 0 \text { and Im}(z) < 0 . \text { So, z lies in the fourth quadrant }. \]
\[ \therefore \arg(z) = - \beta = \alpha - \frac{\pi}{2}\]
\[\text { Thus, z in the polar form is given by }\]
\[z = \sec \alpha \left\{ \cos\left( \alpha - \frac{\pi}{2} \right) + i\sin \left( \alpha - \frac{\pi}{2} \right) \right\} \]
\[\text { Case II }: \]
\[z = \tan \alpha - i \]
\[ \Rightarrow \left| z \right| = \sqrt{\tan^2 + 1}\]
\[ = \left| \sec \alpha \right| \left[ \because \frac{\pi}{2} < \alpha < \pi \right]\]
\[ = - \sec \alpha\]
\[\text { Let } \beta \text { be an acute angle given by } \tan \beta = \left| \frac{Im (z)}{Re(z)} \right|\]
\[\tan \beta = \frac{1}{\left| \tan \alpha \right|}\]
\[ = \left| \cot \alpha \right|\]
\[ = - \cot \alpha\]
\[ = \tan \left( \alpha - \frac{\pi}{2} \right)\]
\[ \Rightarrow \beta = \alpha - \frac{\pi}{2}\]
\[\text{We can see that Re}(z) < 0 \text { and Im} (z) < 0 . So, z \text { lies in the third quadrant }. \]
\[ \therefore \arg(z) = \pi + \beta = \frac{\pi}{2} + \alpha\]
\[\text { Thus, z in the polar form is given by } \]
\[z = - \sec \alpha \left\{ \cos\left( \frac{\pi}{2} + \alpha \right) + i\sin \left( \frac{\pi}{2} + \alpha \right) \right\} \]
APPEARS IN
RELATED QUESTIONS
Express the given complex number in the form a + ib:
`[(1/3 + i 7/3) + (4 + i 1/3)] -(-4/3 + i)`
Express the given complex number in the form a + ib: (1 – i)4
Express the given complex number in the form a + ib: `(1/3 + 3i)^3`
Find the value of the following expression:
\[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\]
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:
\[\frac{(1 - i )^3}{1 - i^3}\]
Express the following complex number in the standard form a + i b:
\[\frac{5 + \sqrt{2}i}{1 - 2\sqrt{i}}\]
Find the real value of x and y, if
\[(3x - 2iy)(2 + i )^2 = 10(1 + i)\]
Find the real value of x and y, if `((1+i)x-2i)/(3+i) + ((2-3i)y+i)/(3-i) = i, xy ∈ R, i = sqrt-1`
Find the smallest positive integer value of m for which \[\frac{(1 + i )^n}{(1 - i )^{n - 2}}\] is a real number.
Evaluate the following:
\[x^4 - 4 x^3 + 4 x^2 + 8x + 44,\text { when } x = 3 + 2i\]
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.
If \[\left| z + 1 \right| = z + 2\left( 1 + i \right)\],find z.
Solve the equation \[\left| z \right| = z + 1 + 2i\].
If π < θ < 2π and z = 1 + cos θ + i sin θ, then write the value of \[\left| z \right|\] .
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 \[\left| z + 4 \right| \leq 3\], then find the greatest and least values of \[\left| z + 1 \right|\].
The polar form of (i25)3 is
If i2 = −1, then the sum i + i2 + i3 +... upto 1000 terms is equal to
The argument of \[\frac{1 - i\sqrt{3}}{1 + i\sqrt{3}}\] is
\[\text { If } z = \frac{1}{(1 - i)(2 + 3i)}, \text { than } \left| z \right| =\]
\[\text { If }z = 1 - \text { cos }\theta + i \text { sin }\theta, \text { then } \left| z \right| =\]
The amplitude of \[\frac{1}{i}\] is equal to
The value of \[(1 + i )^4 + (1 - i )^4\] is
The complex number z which satisfies the condition \[\left| \frac{i + z}{i - z} \right| = 1\] lies on
Simplify : `sqrt(-16) + 3sqrt(-25) + sqrt(-36) - sqrt(-625)`
Find a and b if (a+b) (2 + i) = b + 1 + (10 + 2a)i
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:
`(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:
`(4"i"^8 - 3"i"^9 + 3)/(3"i"^11 - 4"i"^10 - 2)`
Evaluate the following : i116
Show that 1 + i10 + i20 + i30 is a real number
If `((1 - i)/(1 + i))^100` = a + ib, then find (a, b).
Show that `(-1+ sqrt(3)i)^3` is a real number.
