4.3 Trigonometric functions

Trigonometric functions are essential tools for modeling periodic phenomena in math and science. They describe repeating patterns using ratios of right triangle sides and circular motion, allowing us to represent waves, oscillations, and cycles.

These functions form a crucial part of nonlinear models, expanding our ability to analyze complex systems. By understanding their properties and applications, we can tackle a wide range of real-world problems involving periodic behavior and cyclic patterns.

Trigonometric Functions: Definitions and Interpretations

Defining the Six Basic Trigonometric Functions

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• The six basic trigonometric functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot)
• These functions are defined in terms of the ratios of the sides of a right triangle with respect to a given angle
• The sine, cosine, and tangent are the primary trigonometric functions, while the cosecant, secant, and cotangent are their reciprocals

Interpreting Trigonometric Functions in Right Triangles

• In a right triangle, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse
  • For example, in a right triangle with angle θ, $sin(θ) = \frac{opposite}{hypotenuse}$
• The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle
  • For example, in a right triangle with angle θ, $cos(θ) = \frac{adjacent}{hypotenuse}$
• The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side in a right triangle
  • For example, in a right triangle with angle θ, $tan(θ) = \frac{opposite}{adjacent}$
• The cosecant, secant, and cotangent are the reciprocals of the sine, cosine, and tangent, respectively
  • $csc(θ) = \frac{1}{sin(θ)} = \frac{hypotenuse}{opposite}$
  • $sec(θ) = \frac{1}{cos(θ)} = \frac{hypotenuse}{adjacent}$
  • $cot(θ) = \frac{1}{tan(θ)} = \frac{adjacent}{opposite}$

Properties of Trigonometric Functions

Periodicity and Amplitude

• Trigonometric functions are periodic, meaning they repeat their values at regular intervals called periods
  • The sine and cosine functions have a period of 2π radians or 360 degrees, while the tangent function has a period of π radians or 180 degrees
  • For example, $sin(x) = sin(x + 2π)$ and $tan(x) = tan(x + π)$
• The amplitude of a trigonometric function is the maximum absolute value of the function's output, which determines the height of the curve above or below the midline
  • For a function in the form $a \cdot sin(x)$ or $a \cdot cos(x)$, the amplitude is $|a|$
  • For example, the function $2sin(x)$ has an amplitude of 2, while $-3cos(x)$ has an amplitude of 3

Phase Shift and Vertical Shift

• A phase shift is a horizontal translation of a trigonometric function, which moves the curve left or right along the x-axis
  • A phase shift is typically represented by adding or subtracting a constant value within the function's argument, e.g., $sin(x + \frac{π}{4})$ or $cos(x - \frac{π}{3})$
  • A positive phase shift moves the curve to the left, while a negative phase shift moves the curve to the right
• A vertical shift is a vertical translation of a trigonometric function, which moves the curve up or down along the y-axis
  • A vertical shift is typically represented by adding or subtracting a constant value outside the function, e.g., $sin(x) + 2$ or $cos(x) - 1$
  • A positive vertical shift moves the curve up, while a negative vertical shift moves the curve down

Trigonometric Identities and Applications

Fundamental Trigonometric Identities

• Trigonometric identities are equations that are true for all values of the variable for which both sides of the equation are defined
• The Pythagorean identity states that for any angle θ, $sin^2(θ) + cos^2(θ) = 1$
• The reciprocal identities relate the reciprocal trigonometric functions to their counterparts
  • $csc(θ) = \frac{1}{sin(θ)}$, $sec(θ) = \frac{1}{cos(θ)}$, and $cot(θ) = \frac{1}{tan(θ)}$
• The quotient identities express the tangent, cotangent, secant, and cosecant functions in terms of the sine and cosine
  • $tan(θ) = \frac{sin(θ)}{cos(θ)}$ and $cot(θ) = \frac{cos(θ)}{sin(θ)}$
  • $sec(θ) = \frac{1}{cos(θ)}$ and $csc(θ) = \frac{1}{sin(θ)}$

Sum, Difference, and Double-Angle Formulas

• The sum and difference formulas allow for the simplification of expressions involving the sum or difference of two angles
  • $sin(α ± β) = sin(α)cos(β) ± cos(α)sin(β)$
  • $cos(α ± β) = cos(α)cos(β) ∓ sin(α)sin(β)$
• The double-angle formulas express the trigonometric functions of double angles in terms of the functions of the original angle
  • $sin(2θ) = 2sin(θ)cos(θ)$
  • $cos(2θ) = cos^2(θ) - sin^2(θ) = 2cos^2(θ) - 1 = 1 - 2sin^2(θ)$
  • $tan(2θ) = \frac{2tan(θ)}{1 - tan^2(θ)}$

Modeling Periodic Phenomena with Trigonometric Functions

Applications in Physics and Engineering

• Trigonometric functions can be used to model periodic phenomena in various fields, such as physics, engineering, and economics
• Simple harmonic motion, such as the motion of a pendulum or a mass on a spring, can be modeled using sine or cosine functions
  • The displacement, velocity, and acceleration of an object undergoing simple harmonic motion can be described using trigonometric functions with appropriate amplitudes, periods, and phase shifts
  • For example, the displacement of a pendulum can be modeled as $x(t) = Acos(ωt + φ)$, where $A$ is the amplitude, $ω$ is the angular frequency, and $φ$ is the phase shift
• Trigonometric functions can be used to model sound and light waves, describing their amplitude, frequency, and phase
  • For example, a sound wave can be represented as $y(x,t) = Asin(kx - ωt + φ)$, where $A$ is the amplitude, $k$ is the wave number, $ω$ is the angular frequency, and $φ$ is the phase shift

Solving Problems Involving Periodic Phenomena

• In electrical engineering, trigonometric functions are used to represent alternating current (AC) and voltage, with the amplitude and phase of the sine or cosine function corresponding to the peak value and timing of the waveform
  • For example, an AC voltage can be represented as $v(t) = V_psin(ωt + φ)$, where $V_p$ is the peak voltage, $ω$ is the angular frequency, and $φ$ is the phase shift
• Trigonometric functions can be applied to solve problems involving periodic phenomena, such as determining the height of an object at a given time, the time required for a complete oscillation, or the maximum velocity of an object undergoing periodic motion
  • For example, given the equation of motion for a simple pendulum, $x(t) = 0.5cos(πt)$, one can find the height of the pendulum at $t = 0.25$ seconds by evaluating $x(0.25) = 0.5cos(π \cdot 0.25) ≈ 0.35$ meters