The GATE Grind

GATE 2022 AE – Question 13

Engineering Mathematics · Special topics: Fourier series, complex numbers, analytic functions and Cauchy-Riemann equations · 1 mark · Multiple choice

For real x and $i=\sqrt{-1}$, the real part of cos(ix) is

  1. $\sinh x$
  2. $\cosh x$
  3. $\cos x$
  4. $\sin x$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (B) $\cosh x$

Explanation

**Use Euler's formula** $\cos z=\dfrac{e^{iz}+e^{-iz}}{2}$ with $z=ix$:
$$\cos(ix)=\frac{e^{i(ix)}+e^{-i(ix)}}{2}=\frac{e^{-x}+e^{x}}{2}=\cosh x .$$

For real $x$ this is a **real** number, so its real part is the number itself: $\cosh x$ (option B).