Trigonometric Functions: Definitions & Formulas

Trigonometric Functions: Definition

• Considers a circle with a radius of 1, centered at the origin
• Defines $\theta$ as the angle between the x-axis and a point on the circle
• $\cos(\theta)$ refers to the x-coordinate of that point
• $\sin(\theta)$ refers to the y-coordinate of that point

The Unit Circle

• The image shows a unit circle with a radius of 1
• The angle $\theta$ is measured counterclockwise from the positive x-axis
• The x-coordinate of a point on the circle is the cosine ($\cos(\theta)$)
• The y-coordinate of a point on the circle is the sine ($\sin(\theta)$)
• Common angles are labeled in degrees and radians
• Values of cosine and sine are provided for those angles
• Four quadrants of the circle are clearly marked

Trigonometric Functions: Formulas

• Tangent: $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$
• Cosecant: $\csc(\theta) = \frac{1}{\sin(\theta)}$
• Secant: $\sec(\theta) = \frac{1}{\cos(\theta)}$
• Cotangent: $\cot(\theta) = \frac{1}{\tan(\theta)}$
• Pythagorean identity: $\sin^2(\theta) + \cos^2(\theta) = 1$

Trigonometric Functions: Periodicity

• Trigonometric functions exhibit periodicity
• Sine function: period of $2\pi$
• Cosine function: period of $2\pi$
• Tangent function: period of $\pi$

Trigonometric Functions: Transformations

• General form: $f(x) = A\sin(B(x - C)) + D$
• Where:
  • $A$ is the Amplitude
  • $B$ affects the period; Period equals $\frac{2\pi}{B}$
  • $C$ is the phase shift (horizontal translation)
  • $D$ is the vertical shift

Trigonometric Functions: Graph of sin(x)

• Graph displays $y = \sin(x)$
• The x-axis shows radian values ranging from $-2\pi$ to $2\pi$
• The y-axis ranges from -1 to 1
• The sine wave oscillates between -1 and 1
• Key points are marked at $0, \frac{\pi}{2}, \pi, \frac{3\pi}{2},$ and $2\pi$

Trigonometric Functions: Graph of cos(x)

• Graph displays $y = \cos(x)$
• The x-axis shows radian values ranging from $-2\pi$ to $2\pi$
• The y-axis ranges from -1 to 1
• The cosine wave oscillates between -1 and 1
• Key points are marked at $0, \frac{\pi}{2}, \pi, \frac{3\pi}{2},$ and $2\pi$

Trigonometric Functions: Graph of tan(x)

• The graph displays $y = \tan(x)$
• The x-axis shows values in radians
• The y-axis shows the values
• Vertical asymptotes are located at $x = -\frac{3\pi}{2}, -\frac{\pi}{2}, \frac{\pi}{2},$ and $\frac{3\pi}{2}$
• The tangent function approaches but never reaches these asymptotes
• The x-intercepts are located at $x = -\pi, 0,$ and $\pi$

Dynamic Light Scattering: Particle Sizing

Introduction to Dynamic light scattering (DLS)

• DLS, also called PCS or QELS, is used to find the size distribution of particles in a solution
• A laser beam is directed through a colloidal solution, causing particles to scatter light
• A detector picks up the scattered light
• The intensity of the light that scattered light has fluctuations over time because of Brownian motion
• These fluctuations are used to find the particle size

Theory of Dynamic light scattering (DLS)

• Brownian motion happens when particles move randomly in a fluid
• This is caused by collisions with fluid molecules
• Smaller particles move faster
• The diffusion coefficient (D) measures how fast particles diffuse
• The Stokes-Einstein equation expresses its relation to particle size (d): $D = \frac{k_BT}{3\pi\eta d}$
• $k_B$: Boltzmann constant
• $T$: Temperature
• $\eta$: viscosity of the fluid
• Intensity fluctuations of scattered light are due to Brownian motion
• Fluctuation rate correlates with the diffusion coefficient
• The autocorrelation function $g_2(τ)$ measures the fluctuation rate: $g_2(τ) = \frac{}{^2}$
• $I(t)$: scattered light intensity at time t
• $τ$: delay time
• For a monodisperse suspension, the autocorrelation decays exponentially: $g_2(τ) = 1 + β exp(-2Γτ)$
• $β$: constant related to optical setup
• $Γ$: decay rate
• The decay rate relates to the diffusion coefficient by: $$Γ = Dq^2$$
• The scattering vector (q) calculates as: $q = \frac{4\pi n sin(θ/2)}{λ}$
  • $n$: refractive index of the solvent
  • $θ$: scattering angle
  • $λ$: wavelength of the laser light
• By measuring the autocorrelation function, the decay rate, diffusion coefficient, and particle size can be calculated

Data Analysis of Dynamic light scattering (DLS)

• Involves fitting the measured autocorrelation function to the theoretical model to find the decay rate
• For polydisperse suspensions, the autocorrelation combines exponentials: $g_2(τ) = 1 + β[\int G(Γ)exp(-Γτ)dΓ]^2$
• $G(Γ)$: decay rate distribution
• Various algorithms used to invert this for determining size distribution
• A common algorithm is the CONTIN algorithm as a constrained regularization method

Advantages of Dynamic Light Scattering (DLS)

• Non-destructive
• Fast and easy to use
• Can measure particle sizes ranging from nanometers to microns

Disadvantages of Dynamic Light Scattering (DLS)

• Sensitive to contaminants like dust
• Requires knowing the refractive index and viscosity of the solvent
• Assumes particles are spherical

Applications of Dynamic Light Scattering (DLS)

• Characterizing nanoparticles
• Monitoring protein aggregation
• Studying the stability of emulsions

Summary Table: Dynamic Light Scattering (DLS)

• Principle: Measures light intensity fluctuations scattered by particles due to Brownian motion
• Measured Quantity: Autocorrelation function, quantifying intensity correlation over time
• Data Analysis: Determines decay rate from the autocorrelation function, relating this to the diffusion coefficient and particle size
• Stokes-Einstein Equation: Relates diffusion coefficient to hydrodynamic diameter
• Advantages: Non-destructive and relatively fast, suitable for particle sizes in the nanometer to micron range
• Disadvantages: Sensitive to dust, requires solvent properties, assumes spherical shapes
• Applications: Nanoparticle characterization, protein aggregation monitoring, studies on emulsion stability
• Scattering Vector: Defined by equation involving refractive index, scattering angle, and light wavelength
• Decay Rate: Correlates diffusion coefficient with the scattering vector
• Polydisperse Suspension: Describes non-uniform particle sizes, involving decay rate distribution