[ALGEBRA] Trigonometric and geometric applications

This page puts polar form to work in two ways. Powers of $e^{i\theta}$ are easy to compute, so they give identities for sines and cosines. In the Argand plane, conditions on modulus and argument describe regions.

Exponential trigonometry

[PROPOSITION]

Exponential forms of sine and cosine

For $\theta\in\mathbb R$ ,

\cos\theta=\frac{e^{i\theta}+e^{-i\theta}}2, \qquad \sin\theta=\frac{e^{i\theta}-e^{-i\theta}}{2i}.

The two identities follow by adding and subtracting Euler's formula and its conjugate form. They let powers of sine and cosine be expanded with complex arithmetic, where powers of $e^{i\theta}$ combine easily: $(e^{i\theta})^n=e^{in\theta}$ .

[THEOREM]

Binomial theorem over the complex numbers

For $a,b\in\mathbb C$ and every non-negative integer $n$ ,

(a+b)^n=\sum_{k=0}^n{n\choose k}a^{n-k}b^k, \qquad {n\choose k}=\frac{n!}{k!(n-k)!}.

The theorem holds over $\mathbb C$ because complex addition and multiplication obey the same commutative and distributive laws as real arithmetic. The coefficients can be read from Pascal's triangle.

[1]:Complex number
[2]:Expand powers of sine and cosine

[DEFINITION]

Trigonometric polynomial

A trigonometric polynomial is an expression of the form

a_0+\sum_{k=1}^N\bigl(a_k\cos(k\theta)+b_k\sin(k\theta)\bigr), \qquad \theta\in\mathbb R,

where the coefficients $a_k,b_k$ are complex numbers.

[THEOREM]

De Moivre's formula

For $\theta\in\mathbb R$ and $n\in\mathbb Z$ ,

\left(\cos\theta+i\sin\theta\right)^n =\cos(n\theta)+i\sin(n\theta).

More generally, if $z=re^{i\theta}$ with $r>0$ , then

z^n=r^ne^{in\theta}.

[1]:Multiply and divide in polar form
[2]:Calculate a power

[METHOD]

Expand powers of sine and cosine

Replace $\sin\theta$ and $\cos\theta$ by their exponential forms, expand with the binomial theorem and collect conjugate exponential terms back into sines and cosines.

The result is a trigonometric polynomial. The reverse calculation starts with De Moivre's formula and compares real or imaginary parts.

[EXAMPLE]

Expand $\sin^3\theta$

Using the exponential form,

\begin{aligned} \sin^3\theta &=\left(\frac{e^{i\theta}-e^{-i\theta}}{2i}\right)^3\\ &=\frac{e^{3i\theta}-3e^{i\theta}+3e^{-i\theta}-e^{-3i\theta}}{(2i)^3}\\ &=\frac{3\sin\theta-\sin(3\theta)}4. \end{aligned}

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

(\cos\theta+i\sin\theta)^3

gives

\sin(3\theta)=3\cos^2\theta\sin\theta-\sin^3\theta.

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 $3$ .

[routine]

[1]:Expand powers of sine and cosine

Loci in the Argand plane

A locus is the set of points whose coordinates satisfy a stated geometric condition. In the Argand plane, modulus conditions describe distances and argument conditions describe directions.

[METHOD]

Read loci in the Argand plane

Translate a condition on $z$ into a geometric condition. For $a\in\mathbb C$ and $b>0$ ,

|z-a|\le b

is the closed disc with centre $a$ and radius $b$ . For $z\ne a$ and $-\pi<\alpha<\beta\le\pi$ ,

\alpha\le\operatorname{Arg}(z-a)<\beta

is the angular region between the rays from $a$ at arguments $\alpha$ and $\beta$ , including the $\alpha$ boundary and excluding the $\beta$ boundary.

The modulus measures Euclidean distance between points because

|z-w| =\sqrt{\bigl(\operatorname{Re}(z)-\operatorname{Re}(w)\bigr)^2 +\bigl(\operatorname{Im}(z)-\operatorname{Im}(w)\bigr)^2}.

When $z\ne w$ , the argument of $z-w$ is the directed angle made by the line from $w$ to $z$ with the positive real axis. At $z=w$ , the difference is zero and has no argument.

[EXAMPLE]

Interpret a combined locus

Describe

S=\left\{z\in\mathbb C:|z-(1+i)|\le2,\quad 0\le\operatorname{Arg}(z-(1+i))<\frac\pi2\right\}.

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.

[routine]

[1]:Read loci in the Argand plane

Complex polynomials uses the same arithmetic for polynomial coefficients and asks when a polynomial splits into factors.

Trigonometric and geometric applications

De Moivre's formula

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