Mathematics for Physics I - Trigonometric Functions

Basic Definitions & Properties

Trigonometric functions originated in the study of triangles but have evolved into fundamental tools for describing cyclical patterns throughout nature.

Right Triangle Definitions

\[ \sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} \] \[ \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} \] \[ \tan\theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sin\theta}{\cos\theta} \]

These definitions apply directly when θ is an acute angle in a right triangle.

Unit Circle Approach

For any angle θ, we can define trigonometric functions using the unit circle:

The coordinates of the point where the terminal ray intersects the unit circle are \((\cos\theta, \sin\theta)\)

This definition naturally extends trigonometric functions to all angles and reveals their periodic nature:

\[ \sin(\theta + 2\pi) = \sin\theta \] \[ \cos(\theta + 2\pi) = \cos\theta \]

Sine and cosine functions with unit circle representation

Important Identities

Fundamental Identities

\[ \sin^2\theta + \cos^2\theta = 1 \] \[ \tan\theta = \frac{\sin\theta}{\cos\theta} \] \[ \cot\theta = \frac{\cos\theta}{\sin\theta} = \frac{1}{\tan\theta} \] \[ \sec\theta = \frac{1}{\cos\theta} \] \[ \csc\theta = \frac{1}{\sin\theta} \]

Angle Addition Formulas

\[ \sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta \] \[ \cos(\alpha + \beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta \] \[ \tan(\alpha + \beta) = \frac{\tan\alpha + \tan\beta}{1 - \tan\alpha\tan\beta} \]

Double Angle Formulas

\[ \sin(2\theta) = 2\sin\theta\cos\theta \] \[ \cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta \] \[ \tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta} \]

Half Angle Formulas

\[ \sin^2\frac{\theta}{2} = \frac{1 - \cos\theta}{2} \] \[ \cos^2\frac{\theta}{2} = \frac{1 + \cos\theta}{2} \] \[ \tan\frac{\theta}{2} = \frac{1 - \cos\theta}{\sin\theta} = \frac{\sin\theta}{1 + \cos\theta} \]

Euler's Formula

\[ e^{i\theta} = \cos\theta + i\sin\theta \]

Euler's formula provides a powerful connection between trigonometric functions and complex exponentials, simplifying many calculations in physics, particularly in the analysis of waves, AC circuits, and quantum mechanics.

Applications in Physics

Simple Harmonic Motion

The position of a simple harmonic oscillator is described by:

\[ x(t) = A\cos(\omega t + \phi) \]

Where A is the amplitude, ω is the angular frequency, and φ is the phase constant.

The velocity and acceleration are:

\[ v(t) = -A\omega\sin(\omega t + \phi) \] \[ a(t) = -A\omega^2\cos(\omega t + \phi) = -\omega^2 x(t) \]

Wave Phenomena

Traveling waves are described by:

\[ y(x,t) = A\sin(kx - \omega t) \]

Where k is the wave number and ω is the angular frequency.

The wave equation is:

\[ \frac{\partial^2 y}{\partial t^2} = v^2 \frac{\partial^2 y}{\partial x^2} \]

Where v is the wave velocity.

AC Circuits

In alternating current circuits, voltage and current vary as:

\[ V(t) = V_0\sin(\omega t) \] \[ I(t) = I_0\sin(\omega t + \phi) \]

Where φ represents the phase difference between voltage and current.

Using complex notation:

\[ \tilde{V} = V_0 e^{i\omega t} \] \[ \tilde{I} = I_0 e^{i(\omega t + \phi)} \]

Optics

Snell's law of refraction uses the sine function:

\[ \sin\theta_2 = \frac{n_1}{n_2}\sin\theta_1 \]

Where n₁ and n₂ are the refractive indices of the two media.

Diffraction patterns follow:

\[ I(\theta) = I_0 \left(\frac{\sin(\alpha)}{\alpha}\right)^2 \]

Where α depends on the aperture size, wavelength, and angle.

Key Insight:

Trigonometric functions are fundamental to describing periodic phenomena in physics. Their connection to complex exponentials through Euler's formula provides powerful mathematical tools for analyzing oscillatory systems, from simple pendulums to electromagnetic waves and quantum particles.

Trigonometric Series

Trigonometric functions can be represented as infinite series, which are useful in many physics applications:

Taylor Series

\[ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \ldots \] \[ \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \ldots \]

These series are particularly useful for small-angle approximations:

\[ \sin x \approx x \quad \text{for small } x \] \[ \cos x \approx 1 - \frac{x^2}{2} \quad \text{for small } x \]

Fourier Series

Any periodic function f(x) with period 2π can be represented as:

\[ f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left[ a_n \cos(nx) + b_n \sin(nx) \right] \]

Where the coefficients are given by:

\[ a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) \, dx \] \[ b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx) \, dx \]

Fourier series are fundamental in analyzing periodic signals, solving differential equations, and understanding wave phenomena in physics. They allow us to decompose complex periodic functions into simpler sinusoidal components.

Practice Problems

Test your understanding of trigonometric functions with these practice problems:

Prove the identity: \(\sin(A + B)\sin(A - B) = \sin^2 A - \sin^2 B\)
A simple pendulum of length L oscillates with a small amplitude. Show that its period is approximately \(T = 2\pi\sqrt{L/g}\).
Use the small-angle approximation to find the first three non-zero terms in the Taylor series expansion of \(\tan x\).
A wave on a string is described by \(y(x,t) = 0.03 \sin(2x - 4t)\) where x is in meters and t in seconds. Find the wavelength, frequency, period, and wave speed.
Find the Fourier series representation of the square wave function \(f(x) = \begin{cases} 1, & 0 < x < \pi \\ -1, & -\pi < x < 0 \end{cases}\) with period 2π.