Expressing sinnθ and cosnθ in terms of sin(kθ) and cos(kθ)

Multi-angle expressions such as sinnθ and cosnθ involve the trigonometric functions sine and cosine raised to a power or at multiple angles.
The goal is to reduce these to simpler forms expressed in terms of sin(kθ) and cos(kθ), where k is typically a smaller integer than n.

Fundamental to successfully manipulating these forms is De Moivre’s Theorem, which states (cos θ + i sin θ)n = cos nθ + i sin nθ
The theorem, applied to complex numbers, has significant implications for trigonometry and helps us to reduce complex trigonometric expressions to simpler forms.

When n is even, the expression can be simplified using the double angle identities for sin and cos, alongside Pythagoras’ identity sin²θ + cos²θ = 1.
For example, sin²θ can be rewritten as 1 - cos²θ, and cos²θ could in turn be written in terms of cos(2θ) using the double angle identity, leading to considerable simplification.
When n is odd, the expression sinnθ or cosnθ can be manipulated using the identity for sin(θ) cos(2θ) or cos(θ) sin(2θ), reducing it to expressions involving sin(kθ) or cos(kθ) where k<n.

The binomial theorem provides an essential method for expanding expressions when we express sinnθ and cosnθ in terms of sin(kθ) and cos(kθ).
Once De Moivre’s Theorem has been applied to rewrite the expression in binomial form, the Binomial Theorem helps further simplify it.

Beyond pure mathematics, the application of these identities scales up to wave behaviours, notably in physics where it plays an integral part in understanding wave interference and harmonics.
These principles also find use in crystallography in determining symmetry in crystal structures.

Always remember to cross-check and confirm the nature of the trigonometric expression you’re dealing with before deciding on the most appropriate approach.
In exercises, look out for recurring patterns or sequences. These often point towards the use of identities or theorems that can simplify the task at hand.