# Getting value of $\sin x + \cos x$ with complex exponential

I have a bit of difficulty with this. I am trying to express $\sin x + \cos x$ with complex exponential.

I started by using Euler's equations. Then, I used the trigonometric substitution $\sin x = \cos(x+\pi/2)$. The problem is that I always end up with $i - 1$ and $i + 1$ (by using different exponent operations), and the fact $e^{i \pi/2} = 1$ and such.

I'm lost. It would be supposed to give me $\sqrt{2}\cos(X)$, where $X$ is undetermined. I just don't get it.

Could anyone help?

Thanks

• Take $z = e^{ix} + ie^{ix}$, then look at $$\frac{z-\overline{z}}{2i}$$ – Prahlad Vaidyanathan Oct 4 '13 at 19:45

Your first guess (Euler's equation) was good. You have $$e^{ix} = \cos(x) + i \sin(x), \qquad e^{-ix} = \cos(-x) + i \sin(-x) = \cos(x) - i \sin(x)$$ which implies that $$\cos(x) = \frac {2\cos(x)}2 = \frac{\cos(x) + i \sin(x) + \cos(x) - i \sin(x)}2 = \frac{e^{ix} + e^{-ix}}2$$ and $$\sin(x) = \frac {2 i \sin(x)}{2i} = \frac{\cos(x) + i \sin(x) - \cos(x) + i \sin(x)}{2i} = \frac{e^{ix} - e^{-ix}}{2i}.$$ Then you can just add up the expressions and get your exponential formula.

Added : by adding up the previous two equations,

$$\sin(x) + \cos(x) = \frac {(1-i)e^{ix}}2 + \frac{(1+i)e^{-ix}}2 = \frac 1{\sqrt 2} \left( e^{-i\pi/4}e^{ix} + e^{i\pi/4} e^{-ix} \right) = \frac {e^{i(x-\pi/4)} + e^{-i(x-\pi/4)}}{\sqrt 2} = \sqrt 2 \cos(x-\pi/4).$$

Hope that helps,

• I'm sorry, it seems indeed the solution I was about to get, as stated I was ending up with (1 - i) and (1 + i). But I don't get how you got rid of these expressions. I'm sorry if I look like a dumbass, I really don't get it. Seems like you calculated the modulus of (1 - i) and (i + i), giving sqrt(2)/2 = 1/sqrt(2). Hum, can you tell me what you did :( thank you for taking all this time, im grateful – Yannick Oct 4 '13 at 20:37
• Ohhhh nevermind. I got what you did there. Wow... I guess I gotta go bed... Thank you :) – Yannick Oct 4 '13 at 20:46
• @Yannick : Haha, guess I didn't even have to work to help you, just had to wait a bit until you understood yourself! By the way, we've all thought we're a bit stupid every now and then, don't torture yourself over it. Feel free to green-check the answer if you like it. – Patrick Da Silva Oct 4 '13 at 21:22

$$\sin(x)+\cos(x)=\sqrt{2}\left(\frac{1}{\sqrt{2}}\sin (x) + \frac{1}{\sqrt{2}}\cos (x)\right)=\\ \sqrt{2}\left(\sin(\pi/4)\sin (x) + \cos(\pi/4)\cos (x)\right)= \sqrt{2} \cos (x-\pi/4)$$

• That is exactly what the result I'm supposed to get. Problem is... I gotta use complex exponential to achieve the same result. – Yannick Oct 4 '13 at 19:57
• @Yannick : I thought you wanted an expression in terms of the complex exponential? This is just a shifted cosine. – Patrick Da Silva Oct 4 '13 at 19:57
• Yes. The weird thing, the thing I don't get is... The question states "Use complex exponential to find the value of sinx + cosx" and the answer in the solutions is indeed the result found there. What I don't get it how the heck you'd do it with only complex exponential, without making it disgraceful i.e. by "truly" using complex exponential. It's a weird question I must admit. – Yannick Oct 4 '13 at 20:00
• @Yannick : Well look at my answer huh! – Patrick Da Silva Oct 4 '13 at 20:14