Complex Trignometry

Algebra Level 4

If x = t a n ( i ) x=tan(i) , find the value of I m ( x ) Im(x) .

Note:
1) i = − 1 i = \sqrt{-1} .
2) I m ( z ) Im(z) represents the imaginary part of the complex number z z .


The answer is 0.76159415595576488811.

Harsh Depal by Harsh Depal , 23, India

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2 solutions

Harsh Depal
Apr 1, 2014

e i a = c o s ( a ) + i s i n ( a ) ∴ e i a + e − i a = 2 c o s ( a ) A n d e i a − e − i a = 2 i s i n ( a ) i t a n ( a ) = e i a + e − i a e i a − e − i a i t a n ( i ) = e − 1 + e 1 e − 1 − e 1 t a n ( i ) = e 1 + e − 1 e 1 − e − 1 i I m ( x ) = e 1 + e − 1 e 1 − e − 1 I m ( x ) = 0.7615 { e }^{ ia }=cos(a)+isin(a)\\ \therefore { \quad e }^{ ia }+{ e }^{ -ia }=2cos(a)\\ And\\ \quad \quad { e }^{ ia }-{ e }^{ -ia }=2isin(a)\\ i\quad tan(a)=\frac { { e }^{ ia }+{ e }^{ -ia } }{ { e }^{ ia }-{ e }^{ -ia } } \\ i\quad tan(i)=\frac { { e }^{ -1 }+{ e }^{ 1 } }{ { e }^{ -1 }-{ e }^{ 1 } } \\ \\ \quad tan(i)=\frac { { e }^{ 1 }+{ e }^{ -1 } }{ { e }^{ 1 }-{ e }^{ -1 } } \quad i\\ \quad Im(x)=\frac { { e }^{ 1 }+{ e }^{ -1 } }{ { e }^{ 1 }-{ e }^{ -1 } } \\ \quad Im(x)=\boxed { 0.7615 }

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Note: Avoid having text in your Latex. Just place words outside of your equations. I've edited your problem as a reference.

Calvin Lin Staff - 7 years, 2 months ago

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Thanks Calvin,i am new to posting questions and solutions

Harsh Depal - 7 years, 2 months ago

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You can use \cos for cos, \sin for sin, etc.

jatin yadav - 7 years, 2 months ago

Isn't your equation wrong in third step?

Tarunbir Singh - 5 years, 6 months ago

t a n a tan a = c o s a / s i n a cos a/sin a ??

Vishal Yadav - 4 years, 3 months ago
Adib Batous
Mar 7, 2016

we can use Taylor series

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