Question

Find the equations for \(\sin 2 \theta\) and \(\cos 2 \theta\) from the de Moivre equation with \(\mathrm{n}=2\).

Short answer

The equations for \(\sin 2\theta\) and \(\cos 2\theta\) from the de Moivre equation with n=2 are: \[ \sin 2\theta = 2\cos \theta \sin \theta \] \[ \cos 2\theta = \cos^2 \theta - \sin^2 \theta \]

Step-by-step solution

Apply the de Moivre's theorem for n=2

We are given the de Moivre equation as \((\cos \theta + i\sin \theta)^n = \cos n\theta + i\sin n\theta\). We need to find the equations for \(\sin 2\theta\) and \(\cos 2\theta\), so we must set n=2: \[ (\cos \theta + i\sin \theta)^2=\cos 2\theta + i\sin 2\theta \]

Expand the LHS of the equation

To find the expressions for \(\sin 2\theta\) and \(\cos 2\theta\), first, we need to expand the LHS of the equation: \[ (\cos \theta + i\sin \theta)(\cos \theta + i\sin \theta) = \cos^2 \theta + 2i\cos \theta \sin \theta - \sin^2 \theta \] Which simplifies to: \[ (\cos^2 \theta - \sin^2 \theta) + i(2\cos \theta \sin \theta) \]

Equate the real and imaginary parts of the equation

Now, equate the real and imaginary parts of the equation. The real part gives us the expression for \(\cos 2\theta\), and the imaginary part gives us the expression for \(\sin 2\theta\): \[ \cos 2\theta = \cos^2 \theta - \sin^2 \theta \] \[ \sin 2\theta = 2\cos \theta \sin \theta \] Thus, the equations for \(\sin 2\theta\) and \(\cos 2\theta\) are: \[ \sin 2\theta = 2\cos \theta \sin \theta \] \[ \cos 2\theta = \cos^2 \theta - \sin^2 \theta \]

Key concepts

Trigonometric Identities

Trigonometric identities are essential tools in mathematics, used to relate different trigonometric functions to each other. These identities simplify complex expressions and solve equations involving angles.
Understanding and using trigonometric identities can help in many areas, including geometry and calculus. Here are some key aspects:

• Pythagorean Identities: \(\sin^2 \theta + \cos^2 \theta = 1\).
• Sum and Difference Formulas: Used to express trigonometric functions of sums or differences of angles.
• Double Angle Formulas: Such as \(\sin 2\theta = 2 \sin \theta \cos \theta\) and \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta\). These equations can be derived using de Moivre's theorem.

These identities are not just theoretical but have practical applications in physics, engineering, and computer science, where understanding waves and oscillations is crucial.

Complex Numbers

Complex numbers are numbers that have both a real and an imaginary part. They are represented as \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part.
In the context of trigonometry and de Moivre's theorem, complex numbers provide a powerful way to compute powers and roots of numbers.

• Imaginary Unit \(i\): Defined as \(i^2 = -1\).
• Polar Form: Complex numbers can be expressed in polar form as \(r(\cos \theta + i\sin \theta)\), which connects directly to trigonometric functions.
• De Moivre's Theorem: This theorem states that \((\cos \theta + i\sin \theta)^n = \cos n\theta + i\sin n\theta\).

By using de Moivre's theorem, you can calculate expressions involving complex numbers more easily, showing the elegance and interconnectedness of mathematics.

Double Angle Formulas

Double angle formulas are specific trigonometric identities used to find angles that are twice a given angle. They are expressed as \(\sin 2\theta\) and \(\cos 2\theta\), and help in various mathematical computations.
These formulas can be derived using algebra and trigonometry from basic identities:

• \(\sin 2\theta = 2\cos \theta \sin \theta\): Derived using the sum formula \(\sin(a + b) = \sin a \cos b + \cos a \sin b\).
• \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta\): Derived using both the Pythagorean identity and the sum formula \(\cos(a + b) = \cos a \cos b - \sin a \sin b\).
• Applications: Used in calculus, physics, and engineering, particularly in simplifying equations involving harmonic motions.

Understanding these formulas transforms complex equations into manageable forms and highlights the interdependency of different mathematical concepts.