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MATH
[1251]

Methods from Trigonometric and geometric applications

All methods from Trigonometric and geometric applications.

Back to Trigonometric and geometric applications

Expand powers of sine and cosine

Replace sin⁡θ\sin\theta and cos⁡θ\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.

  1. [1]:Exponential forms of sine and cosine
  2. [2]:Binomial theorem over the complex numbers
  3. [3]:Expand sin⁡3θ\sin^3\theta

Read loci in the Argand plane

Translate a condition on zz into a geometric condition. For a∈Ca\in\mathbb C and b>0b>0,

∣z−a∣≤b|z-a|\le b

is the closed disc with centre aa and radius bb. For z≠az\ne a and −π<α<β≤π-\pi<\alpha<\beta\le\pi,

α≤Arg⁡(z−a)<β\alpha\le\operatorname{Arg}(z-a)<\beta

is the angular region between the rays from aa at arguments α\alpha and β\beta, including the α\alpha boundary and excluding the β\beta boundary.

The modulus measures Euclidean distance between points because

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

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

  1. [1]:Argand plane
  2. [2]:Complex modulus
  3. [3]:Argument
  4. [4]:Interpret a combined locus

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Expand powers of sine and cosineRead loci in the Argand plane