Exam Questions - Expressing sin(nθ) and cos(nθ) in terms of sinθ and cosθ

Expressing sin(nθ) and cos(nθ) in terms of sinθ and cosθ

More Complex Trigonometry

The formulae for expressing sin(2θ) in terms of sinθ and cosθ are 2sinθcosθ.
The formulae for expressing cos(2θ) in terms of sinθ and cosθ are cos²θ - sin²θ or 2cos²θ - 1 or 1 - 2sin²θ.
These double angle formulae can be derived from the addition formulae of trigonometric functions.

Sines and Cosine of Multiple Angles

Expressions including sin(nθ) and cos(nθ) can often be transformed into expressions in terms of sinθ and cosθ by repeated applications of the double angle formulae.
Doing so often simplifies the expression and aids in solving problems that rely on these expressions.
As an example, sin(4θ) can be expressed as 2sin(2θ)cos(2θ) using the double angle formulae, and those in turn can be expressed in terms of sinθ and cosθ.

Auxiliary Angle Method

If an expression looks like Rsin(θ + α) or Rcos(θ + α), the method of auxiliary angle can be used.
R is the amplitude, θ is the angle and α is the phase difference.
This method involves expressing the given expression in terms of a single sine or cosine function. The function is easier to handle than the original expression.
The expressions Rsin(θ + α) and Rcos(θ + α) can be expanded using the addition formulae of sine and cosine functions, which gives Rsinθcosα + Rcosθsinα and Rcosθcosα - Rsinθsinα, respectively.

Trigonometric Identities

Occasionally, it may be necessary to use other trigonometric identities to express sin(nθ) and cos(nθ) in terms of sinθ and cosθ, such as the Pythagorean identities or co-function identities.

Tips for Success

Always start by looking for ways to apply the double angle formulae as this will often simplify the expression greatly.
Be aware of the various forms that the double angle formulae can take - they are not limited to just 2sinθcosθ and cos²θ - sin²θ.
Practice a range of problems to strengthen understanding and application techniques of these formulae and identities.