Complex Sine and cosine??

Geometry Level 2

We all know that $\sin^2x+\cos^2x=1$ , but then

$\Large\sin^2(ix)+\cos^2(ix)=?$ where $i$ is the imaginary unit.

-1

Undefined

1

Multivalued

1 solution

Aditya Agarwal
Jan 6, 2016

The complex sine and cosine are defined as: $\sin(ix)=i\sinh(x)$ $\cos(ix)=\cosh(x)$ So $\sin^2(ix)+\cos^2(ix)=\cosh^2x-\sinh^2x=\boxed1$

hey, the question asks for $\sinh^2(ix)+\cosh^2(ix)=-\sin^2(x)+\cos^2(x)=\cos(2x)$ which is multivalued. i think you meant sine and cosine there right? please response i have edited the problem @Aditya Agarwal

Aareyan Manzoor - 5 years, 5 months ago

Oh thank you so much! I am sorry, should I delete and repost the problem? @Aareyan Manzoor

Aditya Agarwal - 5 years, 5 months ago

no need. i will reportg the problem so the staff can mark those who answered "multivaued" correct.

Aareyan Manzoor - 5 years, 5 months ago

Complex Sine and cosine??

1 solution

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