A curious trig identity

Posted on by John

Here is an identity that doesn’t look correct but it is. For real x and y,

|\sin(x + iy)| = |\sin x + \sin iy|

I found the identity in [1]. The author’s proof is short. First of all,

\begin{align*} \sin(x + iy) &= \sin x \cos iy + \cos x \sin iy \\ &= \sin x \cosh y + i \cos x \sinh y \end{align*}

Then

\begin{align*} |\sin(x + iy)|^2 &= \sin^2 x \cosh^2 y + \cos^2 x \sinh^2 y \\ &= \sin^2 x (1 + \sinh^2 y) + (1 -\sin^2x) \sinh^2 y \\ &= \sin^2 x + \sinh^2 y \\ &= |\sin x + i \sinh y|^2 \\ &= |\sin x + \sin iy|^2 \end{align*}

Taking square roots completes the proof.

Now note that the statement at the top assumed x and y are real. You can see that this assumption is necessary by, for example, setting x = 2 and y = i.

Where does the proof use the assumption that x and y are real? Are there weaker assumptions on x and y that are sufficient?

 

[1] R. M. Robinson. A curious trigonometric identity. American Mathematical Monthly. Vol 64, No 2. (Feb. 1957). pp 83–85

4 thoughts on “A curious trig identity”

  1. asada

    The definition of absolute value is where x and y are assumed to be real. For x = 2 and y = i, the expression i \cos x \sinh y is real, not imaginary.

  4. I could not let it go, it really felt like distances…
    I did a bit of research, and came up with this :
    Geometrically:
    ∣ sin (x + iy) ∣ = distance from origin in the plane
    after the coordinate transformation:
    (,) ↦ (sin , sinh )

    The real direction behaves like motion on a circle.
    The imaginary direction behaves like motion on a hyperbola.
    The complex modulus measures Euclidean distance after these two motions combine orthogonally.

    Using the exponential definitions, the identity begins to look like a statement about distances between two exponential points on the complex plane.

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