Advertisements
Advertisements
Question
If π < θ < 2π and z = 1 + cos θ + i sin θ, then write the value of \[\left| z \right|\] .
Advertisements
Solution
\[\pi < \theta < 2\pi\]
\[ \frac{\pi}{2} < \frac{\theta}{2} < \pi \left( \text { Dividing by } 2 \right)\]
\[z = 1 + \cos\theta + i sin\theta\]
\[ \Rightarrow \left| z \right| = \sqrt{\left( 1 + \cos\theta \right)^2 + \sin^2 \theta}\]
\[ \Rightarrow \left| z \right| = \sqrt{1 + \cos^2 \theta + 2\cos\theta + \sin^2 \theta}\]
\[ \Rightarrow \left| z \right| = \sqrt{1 + 1 + 2\cos\theta}\]
\[ \Rightarrow \left| z \right| = \sqrt{2\left( 1 + \cos\theta \right)}\]
\[ \Rightarrow \left| z \right| = \sqrt{2 \times 2 \cos^2 \frac{\theta}{2}}\]
\[ \Rightarrow \left| z \right| = 2\sqrt{\cos^2 \frac{\theta}{2}}\]
\[ \Rightarrow \left| z \right| = - 2\cos\frac{\theta}{2} \left[ \text { Since } \frac{\pi}{2} < \frac{\theta}{2} < \pi , \cos\frac{\theta}{2} \text { is negative } \right]\]
APPEARS IN
RELATED QUESTIONS
Express the given complex number in the form a + ib: i–39
Let z1 = 2 – i, z2 = –2 + i. Find Re`((z_1z_2)/barz_1)`
Evaluate the following:
\[\frac{1}{i^{58}}\]
Evaluate the following:
\[i^{37} + \frac{1}{i^{67}}\].
Evaluate the following:
\[( i^{77} + i^{70} + i^{87} + i^{414} )^3\]
Find the value of the following expression:
i49 + i68 + i89 + i110
Express the following complex number in the standard form a + i b:
\[(1 + 2i )^{- 3}\]
Find the real value of x and y, if
\[(3x - 2iy)(2 + i )^2 = 10(1 + i)\]
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
Im `(1/(z_1overlinez_1))`
Find the smallest positive integer value of m for which \[\frac{(1 + i )^n}{(1 - i )^{n - 2}}\] is a real number.
If \[a = \cos\theta + i\sin\theta\], find the value of \[\frac{1 + a}{1 - a}\].
Evaluate the following:
\[x^4 + 4 x^3 + 6 x^2 + 4x + 9, \text { when } x = - 1 + i\sqrt{2}\]
Evaluate the following:
\[2 x^4 + 5 x^3 + 7 x^2 - x + 41, \text { when } x = - 2 - \sqrt{3}i\]
Solve the system of equations \[\text { Re }\left( z^2 \right) = 0, \left| z \right| = 2\].
If \[\left| z + 1 \right| = z + 2\left( 1 + i \right)\],find z.
Solve the equation \[\left| z \right| = z + 1 + 2i\].
What is the smallest positive integer n for which \[\left( 1 + i \right)^{2n} = \left( 1 - i \right)^{2n}\] ?
Find the number of solutions of \[z^2 + \left| z \right|^2 = 0\].
If n is any positive integer, write the value of \[\frac{i^{4n + 1} - i^{4n - 1}}{2}\].
If \[\frac{\left( a^2 + 1 \right)^2}{2a - i} = x + iy\] find the value of \[x^2 + y^2\].
Write the sum of the series \[i + i^2 + i^3 + . . . .\] upto 1000 terms.
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 `(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 a = cos θ + i sin θ, then \[\frac{1 + a}{1 - a} =\]
If \[z = \left( \frac{1 + i}{1 - i} \right)\] then z4 equals
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:
`(3 + 2"i")/(2 - 5"i") + (3 -2"i")/(2 + 5"i")`
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)`
If a = cosθ + isinθ, find the value of `(1 + "a")/(1 - "a")`.
Show that `(-1 + sqrt3 "i")^3` is a real number.
Find the value of `(i^(592) + i^(590) + i^(588) + i^(586) + i^(584))/(i^(582) + i^(580) + i^(578) + i^(576) + i^(574))`
Show that `(-1+sqrt3i)^3` is a real number.
