Sine, Cosine, and Tangent Functions

Introduction

Key Concepts

1. Definitions and Basic Properties

Sine, cosine, and tangent are the primary trigonometric functions derived from the ratios of sides in a right-angled triangle. Given an angle θ in a right-angled triangle:

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

These functions are periodic, with sine and cosine having a period of $2\pi$ radians and tangent having a period of $\pi$ radians. Understanding their basic properties is crucial for graphing and solving trigonometric equations.

2. The Unit Circle

The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. It serves as a foundational tool for defining trigonometric functions for all real numbers.

For any angle θ, measured in radians from the positive x-axis, the coordinates $(x, y)$ of the corresponding point on the unit circle are:

• $x = \cos(\theta)$
• $y = \sin(\theta)$

This representation allows for the extension of sine and cosine functions beyond acute angles, facilitating the analysis of trigonometric functions in various quadrants.

3. Graphs of Sine, Cosine, and Tangent Functions

Understanding the graphical representations of trigonometric functions is essential for analyzing their behavior and solving related equations.

• Graph of Sine: $ y = \sin(\theta) $ oscillates between -1 and 1, with a period of $2\pi$, amplitude of 1, and phase shift of 0.
• Graph of Cosine: $ y = \cos(\theta) $ also oscillates between -1 and 1, sharing the same period and amplitude as the sine function but is phase-shifted by $\frac{\pi}{2}$ radians.
• Graph of Tangent: $ y = \tan(\theta) $ has vertical asymptotes at $\theta = \frac{\pi}{2} + k\pi$, where $k$ is an integer. It repeats every $\pi$ radians and ranges from $-\infty$ to $\infty$.

4. Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all permissible values of the variable. They are fundamental 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(a \pm b) = \sin(a)\cos(b) \pm \cos(a)\sin(b) $
  • $ \cos(a \pm b) = \cos(a)\cos(b) \mp \sin(a)\sin(b) $
  • $ \tan(a \pm b) = \frac{\tan(a) \pm \tan(b)}{1 \mp \tan(a)\tan(b)} $

5. Inverse Trigonometric Functions

Inverse trigonometric functions allow for the determination of an angle given a trigonometric ratio. They are essential for solving equations where the angle is the unknown.

• $ \sin^{-1}(x) = \theta \Rightarrow \sin(\theta) = x $
• $ \cos^{-1}(x) = \theta \Rightarrow \cos(\theta) = x $
• $ \tan^{-1}(x) = \theta \Rightarrow \tan(\theta) = x $

6. Applications of Trigonometric Functions

Trigonometric functions are widely used in various applications, including:

• Engineering: Analyzing waveforms, signal processing.
• Physics: Describing oscillatory motion, such as pendulums and springs.
• Architecture: Designing structures with specific angles and dimensions.
• Navigation: Calculating distances and angles in mapping and GPS technologies.

Advanced Concepts

1. Trigonometric Equations and Their Solutions

Solving trigonometric equations involves finding all angles that satisfy a given trigonometric identity within a specified interval. Techniques include:

• Using identities to simplify the equation.
• Applying inverse trigonometric functions to isolate the variable.
• Considering all possible solutions within the period of the function.

For example, solving $ \sin(\theta) = \frac{1}{2} $ within $0 \leq \theta < 2\pi$ yields $ \theta = \frac{\pi}{6} $ and $ \theta = \frac{5\pi}{6} $.

2. Graph Transformations

Understanding how different transformations affect the graphs of trigonometric functions is essential for modeling and analysis.

• Amplitude Change: $ y = A\sin(\theta) $ or $ y = A\cos(\theta) $ alters the vertical stretch/compression.
• Frequency and Period Adjustments: $ y = \sin(B\theta) $ changes the period to $ \frac{2\pi}{B} $.
• Phase Shifts: $ y = \sin(\theta - C) $ shifts the graph horizontally by C units.
• Vertical Shifts: $ y = \sin(\theta) + D $ moves the graph vertically by D units.

These transformations allow for the customization of trigonometric models to fit specific scenarios.

3. Fourier Series

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

A Fourier series for a function f(θ) is given by: $$ f(\theta) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos(n\theta) + b_n \sin(n\theta) \right) $$

where the coefficients $ a_n $ and $ b_n $ are determined based on the function's properties.

4. Complex Numbers and Euler's Formula

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

This relationship is pivotal in advanced mathematics and engineering, facilitating the analysis of oscillatory systems and waveforms.

5. Applications in Calculus

Trigonometric functions play a critical role in calculus, particularly in differentiation and integration.

• Differentiation:
  • $ \frac{d}{d\theta} \sin(\theta) = \cos(\theta) $
  • $ \frac{d}{d\theta} \cos(\theta) = -\sin(\theta) $
  • $ \frac{d}{d\theta} \tan(\theta) = \sec^2(\theta) $
• Integration:
  • $ \int \sin(\theta) d\theta = -\cos(\theta) + C $
  • $ \int \cos(\theta) d\theta = \sin(\theta) + C $
  • $ \int \tan(\theta) d\theta = -\ln|\cos(\theta)| + C $

These operations are essential for solving problems involving rates of change and areas under curves.

6. Trigonometric Substitutions in Integration

Trigonometric substitutions simplify the integration of rational functions by introducing trigonometric identities to replace algebraic expressions. Common substitutions include:

• $ x = a\sin(\theta) $ for integrals involving $ \sqrt{a^2 - x^2} $
• $ x = a\tan(\theta) $ for integrals involving $ \sqrt{a^2 + x^2} $
• $ x = a\sec(\theta) $ for integrals involving $ \sqrt{x^2 - a^2} $

These substitutions transform the integral into a trigonometric form, which is often easier to evaluate.

7. Inverse Trigonometric Functions and Their Applications

Inverse trigonometric functions are used to solve equations involving trigonometric ratios and to describe angles based on given ratios. They are essential in various applications, including:

• Determining angles in triangles when certain sides are known.
• Modeling oscillatory behavior in physics and engineering.
• Solving integrals and differential equations in calculus.

Comparison Table

Summary and Key Takeaways