Examples from Trigonometric and geometric applications

Expand $\sin^3\theta$

Using the exponential form,

For the last step, group opposite exponents and use $e^{ik\theta}-e^{-ik\theta}=2i\sin(k\theta)$, together with $(2i)^3=-8i$. This step turns the exponential calculation back into a real trigonometric identity.

For the reverse direction, the imaginary part of

gives

Using $\cos^2\theta=1-\sin^2\theta$ recovers the same cubic identity.

Either way, the cube of one oscillation becomes a combination of oscillations at frequencies $1$ and $3$.

Interpret a combined locus

Describe

Writing $w=z-(1+i)$ moves the centre to the origin. The modulus condition restricts $w$ to the closed disc of radius $2$; the argument condition keeps only directions from the positive real axis up to, but not including, the positive imaginary axis. Thus $S$ is the quarter-disc of radius $2$ lying above and to the right of $1+i$, with its boundary only partly included:

• the horizontal radial edge is included, except for the centre;
• the circular arc is included, except for its upper endpoint;
• the vertical radial edge is excluded.

The centre is omitted because $\operatorname{Arg}(0)$ is undefined, a detail the disc condition alone would miss.