Advertisements
Advertisements
प्रश्न
If a = cosθ + isinθ, find the value of `(1 + "a")/(1 - "a")`.
उत्तर
Given that: a = cosθ + isinθ
∴ `(1 + a)/(1 - a) = (1 + cos theta + i sin theta)/(1 - cos theta - i sin theta)`
= `(1 + cos theta + i sin theta)/(1 - cos theta - i sin theta) xx (1 - cos theta + i sin theta)/(1 - cos theta + i sin theta)`
= `(1 - cos theta + i sin theta + cos theta - cos^2 theta + i sin theta cos theta + i sin theta - i sin theta cos theta + i^2 sin^2 theta)/((1 - cos theta)^2 - i^2 sin^2 theta)`
= `(1 + i sin theta - cos^2 theta + i sin theta - sin^2 theta)/(1 + cos^2 theta - 2 cos theta + sin^2 theta)`
= `(sin^2 theta + 2i sin theta - sin^2 theta)/(1 + 1 - 2 cos theta)`
= `(2i sin theta)/(2 - 2 cos theta)`
= `(2i sin theta)/(2(1 - cos theta))`
= `(i sin theta)/(1 - cos theta)`
= `(2 sin theta/2 cos theta/2.i)/(2sin^2 theta/2)`
= `cot theta/2 . i`
Hence, `(1 + a)/(1 - a) = icot theta/2`.
APPEARS IN
संबंधित प्रश्न
Evaluate the following:
i457
Evaluate the following:
\[i^{49} + i^{68} + i^{89} + i^{110}\]
Express the following complex number in the standard form a + i b:
\[\frac{1 - i}{1 + i}\]
Express the following complex number in the standard form a + i b:
\[\frac{(2 + i )^3}{2 + 3i}\]
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:
\[\frac{3 - 4i}{(4 - 2i)(1 + i)}\]
Find the real value of x and y, if
\[(3x - 2iy)(2 + i )^2 = 10(1 + i)\]
Find the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
Find the smallest positive integer value of m for which \[\frac{(1 + i )^n}{(1 - i )^{n - 2}}\] is a real number.
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 \[z = - i \bar{z}\].
Solve the system of equations \[\text { Re }\left( z^2 \right) = 0, \left| z \right| = 2\].
If z1, z2, z3 are complex numbers such that \[\left| z_1 \right| = \left| z_2 \right| = \left| z_3 \right| = \left| \frac{1}{z_1} + \frac{1}{z_2} + \frac{1}{z_3} \right| = 1\] then find the value of \[\left| z_1 + z_2 + z_3 \right|\] .
Write (i25)3 in polar form.
Express the following complex in the form r(cos θ + i sin θ):
\[\frac{1 - i}{\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}}\]
Write the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] 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 i2 = −1, then the sum i + i2 + i3 +... upto 1000 terms is equal to
If z is a non-zero complex number, then \[\left| \frac{\left| z \right|^2}{zz} \right|\] is equal to
\[(\sqrt{- 2})(\sqrt{- 3})\] is equal to
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 z is a complex number, then
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 + 3i)(2 – 3i)
Show that `(-1 + sqrt(3)"i")^3` is a real number
Find the value of `(3 + 2/"i")("i"^6 - "i"^7)(1 + "i"^11)`
Evaluate the following : i888
If z1 = 3 – 2i and z2 = –1 + 3i, then Im(z1z2) = ______.
Show that `(-1+sqrt3i)^3` is a real number.
