Sine, Cosine, and Tangent Functions

Introduction

Key Concepts

Definitions and Basic Properties

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. The primary trigonometric functions—sine ($\sin$), cosine ($\cos$), and tangent ($\tan$)—are defined based on a right-angled triangle. Consider a right-angled triangle with an angle $\theta$:

• Sine: $\sin(\theta) = \frac{\text{Opposite side}}{\text{Hypotenuse}}$
• Cosine: $\cos(\theta) = \frac{\text{Adjacent side}}{\text{Hypotenuse}}$
• Tangent: $\tan(\theta) = \frac{\text{Opposite side}}{\text{Adjacent side}} = \frac{\sin(\theta)}{\cos(\theta)}$

These definitions establish the foundational relationships that are critical in various applications, such as determining unknown sides or angles in triangles.

The Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It serves as a fundamental tool in defining trigonometric functions for all real numbers, extending their applicability beyond acute angles in right-angled triangles.

• Sine: For any angle $\theta$, $\sin(\theta)$ is the y-coordinate of the point on the unit circle.
• Cosine: $\cos(\theta)$ is the x-coordinate of the same point.
• Tangent: $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$, representing the slope of the line formed by the angle $\theta$.

The unit circle facilitates the exploration of periodicity, amplitude, and phase shifts in trigonometric functions.

Graphs of Trigonometric Functions

Understanding the graphical representations of sine, cosine, and tangent functions is crucial for visualizing their behavior and properties.

• Sine Function ($\sin(\theta)$): A periodic function with a period of $2\pi$, amplitude of 1, and it oscillates between -1 and 1.
• Cosine Function ($\cos(\theta)$): Similar to the sine function but shifted $\frac{\pi}{2}$ radians to the left.
• Tangent Function ($\tan(\theta)$): A periodic function with a period of $\pi$, no amplitude restrictions, and vertical asymptotes where $\cos(\theta) = 0$.

Key Identities and Formulas

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables. They are essential for simplifying expressions and solving equations.

• Pythagorean Identity: $\sin^2(\theta) + \cos^2(\theta) = 1$
• Reciprocal Identities:
  • $\csc(\theta) = \frac{1}{\sin(\theta)}$
  • $\sec(\theta) = \frac{1}{\cos(\theta)}$
  • $\cot(\theta) = \frac{1}{\tan(\theta)}$
• Angle Sum and Difference Identities:
  • $\sin(\alpha \pm \beta) = \sin(\alpha)\cos(\beta) \pm \cos(\alpha)\sin(\beta)$
  • $\cos(\alpha \pm \beta) = \cos(\alpha)\cos(\beta) \mp \sin(\alpha)\sin(\beta)$

Inverse Trigonometric Functions

Inverse trigonometric functions allow for the determination of angles when the values of sine, cosine, or tangent are known. They are essential for solving equations where the trigonometric functions are involved.

• Inverse Sine: $\sin^{-1}(x)$
• Inverse Cosine: $\cos^{-1}(x)$
• Inverse Tangent: $\tan^{-1}(x)$

Applications of Trigonometric Functions

Trigonometric functions have extensive applications in various fields:

• Physics: Describing wave motion, oscillations, and harmonic motion.
• Engineering: Analyzing forces, designing structures, and signal processing.
• Computer Graphics: Modeling periodic phenomena and creating animations.
• Navigation: Calculating distances and plotting courses using angles.

Solving Trigonometric Equations

Solving trigonometric equations involves finding all angles that satisfy a given trigonometric equation within a specified interval. Techniques include using identities, algebraic manipulation, and inverse functions.

For example, to solve $\sin(\theta) = \frac{1}{2}$ for $0 \leq \theta < 2\pi$:

1. Identify the reference angle: $\theta = \frac{\pi}{6}$.
2. Determine all possible solutions within the interval: $\theta = \frac{\pi}{6}, \frac{5\pi}{6}$.

Periodic Nature and Amplitude

Trigonometric functions are periodic, meaning they repeat their values in regular intervals:

• Sine and Cosine: Period of $2\pi$ radians.
• Tangent: Period of $\pi$ radians.

The amplitude refers to the maximum value the function attains. For sine and cosine functions, the amplitude is 1.

Phase Shift and Vertical Shift

Phase shift refers to the horizontal displacement of the function, while vertical shift refers to its vertical displacement. These shifts alter the position of the trigonometric function's graph without affecting its shape.

• Phase Shift: Occurs when the function includes a term like $\sin(\theta - c)$ or $\cos(\theta + c)$.
• Vertical Shift: Occurs when the function includes an added constant, such as $\sin(\theta) + d$.

Amplitude Modulation and Frequency

Amplitude modulation involves changing the amplitude of the trigonometric function, while frequency refers to the number of cycles the function completes in a unit interval.

• Amplitude: Affects the height of the peaks and troughs of the graph.
• Frequency: Determines how closely packed the waves are on the graph.

Advanced Concepts

Derivation of Trigonometric Functions from the Unit Circle

The unit circle provides a geometric interpretation of trigonometric functions for all real numbers. By extending the definitions beyond acute angles, functions such as sine and cosine are defined for any real angle $\theta$.

Consider an angle $\theta$ in standard position. The coordinates of the point where the terminal side intersects the unit circle are $(\cos(\theta), \sin(\theta))$. This definition ensures that trigonometric functions are inherently periodic and extends their applicability to both positive and negative angles.

The tangent function can be derived as the slope of the line formed by the angle $\theta$, resulting in $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$. This relationship is fundamental in deriving other identities and solving complex trigonometric equations.

Proving Fundamental Identities

Proving trigonometric identities requires a solid grasp of the fundamental properties and relationships between trigonometric functions. One essential identity is the Pythagorean Identity:

This can be derived from the Pythagorean theorem applied to the unit circle:

1. In the unit circle, the hypotenuse is 1, so $x^2 + y^2 = 1$, where $x = \cos(\theta)$ and $y = \sin(\theta)$.
2. Substituting, we get $\cos^2(\theta) + \sin^2(\theta) = 1$.

This identity is the cornerstone for deriving other identities, such as the double-angle and half-angle formulas.

Double-Angle and Half-Angle Formulas

Double-angle formulas express trigonometric functions of double angles ($2\theta$) in terms of single angles ($\theta$). They are derived by applying angle sum identities with $\alpha = \beta = \theta$.

• $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$
• $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) = 2\cos^2(\theta) - 1 = 1 - 2\sin^2(\theta)$
• $\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}$

Half-angle formulas are similarly derived and are useful in integrating trigonometric expressions and solving complex equations.

Trigonometric Equations and Their Solutions

Solving trigonometric equations often involves multiple steps, including applying identities, factoring, and using inverse functions. For example, to solve the equation $\sin(\theta) = \sin(\alpha)$ for $0 \leq \theta < 2\pi$:

1. Use the identity: $\theta = \alpha + 2k\pi$ or $\theta = \pi - \alpha + 2k\pi$ for any integer $k$.
2. Within the interval $0 \leq \theta < 2\pi$, the solutions are $\theta = \alpha$ and $\theta = \pi - \alpha$.

Applications in Fourier Series

Fourier series decompose periodic functions into sums of sine and cosine functions. This decomposition is fundamental in fields like signal processing, acoustics, and electrical engineering.

A Fourier series for a function $f(x)$ is given by: $$ f(x) = a_0 + \sum_{n=1}^{\infty} [a_n \cos(n\omega x) + b_n \sin(n\omega x)] $$

where $a_0$, $a_n$, and $b_n$ are coefficients determined by the function, and $\omega$ is the fundamental frequency.

Complex Numbers and Euler's Formula

Euler's formula establishes a deep connection between trigonometric functions and complex exponentials: $$ e^{i\theta} = \cos(\theta) + i\sin(\theta) $$

This relationship is crucial in various areas of engineering and physics, including electrical engineering and quantum mechanics. It allows for the simplification of complex calculations involving oscillations and waves.

Trigonometric Integrals and Differentiation

Differentiation and integration of trigonometric functions are fundamental skills in calculus, essential for solving problems in physics, engineering, and other applied sciences.

• Derivatives:
  • $\frac{d}{dx} \sin(x) = \cos(x)$
  • $\frac{d}{dx} \cos(x) = -\sin(x)$
  • $\frac{d}{dx} \tan(x) = \sec^2(x)$
• Integrals:
  • $\int \sin(x) \, dx = -\cos(x) + C$
  • $\int \cos(x) \, dx = \sin(x) + C$
  • $\int \tan(x) \, dx = -\ln|\cos(x)| + C$

Trigonometric Function Transformations

Transformations of trigonometric functions involve shifting, scaling, and reflecting their graphs. These transformations are useful in modeling real-world phenomena where amplitude, frequency, or phase changes occur.

• Amplitude Change: $A\sin(\theta)$ scales the sine function vertically by a factor of $A$.
• Frequency Shift: $\sin(B\theta)$ alters the period of the sine function to $\frac{2\pi}{B}$.
• Phase Shift: $\sin(\theta - C)$ shifts the graph horizontally by $C$ units.

Interdisciplinary Connections

Trigonometric functions intersect with numerous disciplines:

• Physics: Analyzing oscillatory motion, waves, and circular motion.
• Engineering: Designing electrical circuits, signal processing, and structural analysis.
• Computer Science: Computer graphics, algorithm design, and cryptography.
• Economics: Modeling periodic trends and cycles in financial markets.

Advanced Problem Solving

Complex problems often require combining multiple trigonometric concepts and identities. For instance, solving integrals involving trigonometric functions may necessitate the use of substitution, integration by parts, or trigonometric identities to simplify the expression.

Example Problem:

1. Evaluate $\int \sin^2(\theta) \, d\theta$.
2. Use the identity $\sin^2(\theta) = \frac{1 - \cos(2\theta)}{2}$.
3. Integrate: $\int \frac{1 - \cos(2\theta)}{2} \, d\theta = \frac{\theta}{2} - \frac{\sin(2\theta)}{4} + C$.

Trigonometric Series and Convergence

Understanding the convergence of trigonometric series is essential in both pure and applied mathematics. Analyzing the conditions under which a series converges involves exploring concepts like uniform convergence and absolute convergence.

For example, the convergence of the Fourier series depends on the smoothness and periodicity of the function being represented.

Trigonometric Substitutions in Integration

Trigonometric substitutions are techniques used to simplify integrals involving square roots of quadratic expressions. By substituting a trigonometric function for a variable, the integral becomes more manageable.

Example:

1. Evaluate $\int \frac{dx}{\sqrt{a^2 - x^2}}$.
2. Let $x = a\sin(\theta)$, then $dx = a\cos(\theta) d\theta$.
3. Substitute: $\int \frac{a\cos(\theta) d\theta}{\sqrt{a^2 - a^2\sin^2(\theta)}} = \int \frac{a\cos(\theta) d\theta}{a\cos(\theta)} = \int d\theta = \theta + C$.
4. Convert back: $\theta = \sin^{-1}\left(\frac{x}{a}\right)$.
5. Final Solution: $\sin^{-1}\left(\frac{x}{a}\right) + C$.

Trigonometric Function Applications in Differential Equations

Trigonometric functions frequently appear in solutions to differential equations, particularly those modeling oscillatory systems like springs and electrical circuits.

Example:

1. Consider the differential equation $\frac{d^2y}{dx^2} + y = 0$.
2. Characteristic equation: $r^2 + 1 = 0 \rightarrow r = \pm i$.
3. General Solution: $y(x) = C_1 \cos(x) + C_2 \sin(x)$.

Comparison Table

Summary and Key Takeaways