Advertisements
Advertisements
प्रश्न
If \[a = \cos\theta + i\sin\theta\], find the value of \[\frac{1 + a}{1 - a}\].
Advertisements
उत्तर
\[\frac{1 + a}{1 - a} = \frac{1 + \cos\theta + i\sin\theta}{1 - \cos\theta - i\sin\theta}\]
\[ = \frac{(1 + cos\theta) + isin\theta}{(1 - cos\theta) - isin\theta} \times \frac{(1 - cos\theta) + isin\theta}{(1 - cos\theta) + isin\theta}\]
\[ = \frac{1 - \cos\theta + i\sin\theta + \cos\theta - \cos^2 \theta + i\cos\theta\sin\theta + i\sin\theta - i\sin\theta\cos\theta + i^2 \sin^2 \theta}{(1 - \cos\theta )^2 - i^2 \sin^2 \theta}\]
\[ = \frac{1 - \cos^2 \theta - \sin^2 \theta + 2i\sin\theta}{1 + \cos^2 \theta - 2i\cos\theta + \sin^2 \theta}\phantom{.....}...[ \because i^2 = - 1]\]
\[ = \frac{\sin^2 \theta - \sin^2 \theta + 2i\sin\theta}{2 - 2i\cos\theta} \phantom{.....}...[ \because \cos^2 \theta + \sin^2 \theta = 1]\]
\[ = \frac{i\sin\theta}{1 - \cos\theta}\]
\[ = \frac{\cancel{2}i\cancel{\sin\frac{\theta}{2}}\cos\frac{\theta}{2}}{\cancel{2}\cancel{\sin^2 \frac{\theta}{2}}}\]
\[ = i\cot\frac{\theta}{2}\]
Thus, \[\frac{1 + a}{1 - a} = i\cot\frac{\theta}{2}\].
APPEARS IN
संबंधित प्रश्न
Let z1 = 2 – i, z2 = –2 + i. Find Re`((z_1z_2)/barz_1)`
Let z1 = 2 – i, z2 = –2 + i. Find `"Im"(1/(z_1barz_1))`
Evaluate the following:
(ii) i528
Evaluate the following:
\[i^{30} + i^{40} + i^{60}\]
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:
\[\frac{1}{(2 + i )^2}\]
Express the following complex number in the standard form a + ib:
\[\frac{(2 + i )^3}{2 + 3i}\]
Express the following complex number in the standard form a + i b:
\[\frac{2 + 3i}{4 + 5i}\]
If \[z_1 = 2 - i, z_2 = 1 + i,\text { find } \left| \frac{z_1 + z_2 + 1}{z_1 - z_2 + i} \right|\]
If \[z_1 = 2 - i, z_2 = - 2 + i,\] find
Re \[\left( \frac{z_1 z_2}{z_1} \right)\]
Find the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
Evaluate the following:
\[2 x^3 + 2 x^2 - 7x + 72, \text { when } x = \frac{3 - 5i}{2}\]
Evaluate the following:
\[x^4 + 4 x^3 + 6 x^2 + 4x + 9, \text { when } x = - 1 + i\sqrt{2}\]
For a positive integer n, find the value of \[(1 - i )^n \left( 1 - \frac{1}{i} \right)^n\].
If \[\left| z + 1 \right| = z + 2\left( 1 + i \right)\],find z.
Find the number of solutions of \[z^2 + \left| z \right|^2 = 0\].
If \[\frac{\left( a^2 + 1 \right)^2}{2a - i} = x + iy\] find the value of \[x^2 + y^2\].
Write the value of \[\sqrt{- 25} \times \sqrt{- 9}\].
Write the value of \[\arg\left( z \right) + \arg\left( \bar{z} \right)\].
Find the real value of a for which \[3 i^3 - 2a i^2 + (1 - a)i + 5\] is real.
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 \[\left( 1 + \sqrt{3} \right)\].
If\[z = \cos\frac{\pi}{4} + i \sin\frac{\pi}{6}\], then
The polar form of (i25)3 is
If i2 = −1, then the sum i + i2 + i3 +... upto 1000 terms is equal to
If a = cos θ + i sin θ, then \[\frac{1 + a}{1 - a} =\]
\[\text { If } z = \frac{1}{(2 + 3i )^2}, \text { than } \left| z \right| =\]
\[\text { If }z = 1 - \text { cos }\theta + i \text { sin }\theta, \text { then } \left| z \right| =\]
If \[z = \frac{1}{1 - cos\theta - i sin\theta}\] then Re (z) =
The amplitude of \[\frac{1}{i}\] is equal to
If \[f\left( z \right) = \frac{7 - z}{1 - z^2}\] , where \[z = 1 + 2i\] then \[\left| f\left( z \right) \right|\] is
The complex number z which satisfies the condition \[\left| \frac{i + z}{i - z} \right| = 1\] lies on
If z is a complex number, then
Which of the following is correct for any two complex numbers z1 and z2?
If the complex number \[z = x + iy\] satisfies the condition \[\left| z + 1 \right| = 1\], then z lies on
Evaluate the following : i403
Evaluate the following : `1/"i"^58`
