×

# Help Needed!

If a=$$\frac { 1 }{ 4 } +i\frac { \sqrt { 3 } }{ 4 }$$ and $$z=x+iy$$, then $$\sin ^{ -1 }{ { \left| z \right| }^{ 2 } } +\cos ^{ -1 }{ (a\bar { z } } +\bar { a } z-2)$$ equals to,

$$(A)0$$

$$(B)\frac { \pi }{ 4 }$$

$$(C)\frac { \pi }{ 2 }$$

$$(D)\frac { 3\pi }{ 2 }$$

Note by Anandhu Raj
2 years, 4 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

Sort by:

We can write a and z as follows $a=\frac {1}{2}e^{\frac {i \pi}{3}}$ $z=|z|e^{i\theta }$

$a \overline{z}+z \overline{a}-2 =\frac {|z|}{2} (e^{(-i )\frac {\pi}{3} }e^{i \theta} + e^{(i \frac {\pi}{3}} e^{(-i )\theta})-2$

$=|z| \cos (\theta-\frac {\pi}{3})-2$

Now using domain of the inverse functions

$-1\leq |z|^{2} \leq 1$ $-1 \leq |z| \cos (\theta-\frac {\pi}{3})-2 \leq 1$

From these it is easy to see that $|z|= 1 and \cos (\theta-\frac {\pi}{3})=1$

So we get the answer as $sin^{-1}(1)+cos^{-1}(-1)= 3\pi /2$which is option D. Enjoy.

- 2 years, 4 months ago

Thank you :)

- 2 years, 4 months ago