Sine, Cosine, and Tangent Functions

Introduction

Key Concepts

1. Definitions and Basic Properties

Trigonometric functions relate the angles of a triangle to the lengths of its sides. The primary functions—sine, cosine, and tangent—are defined based on a right-angled triangle or the unit circle.

In a right-angled triangle:

• Sine ($\sin$): The ratio of the length of the opposite side to the hypotenuse.
• Cosine ($\cos$): The ratio of the length of the adjacent side to the hypotenuse.
• Tangent ($\tan$): The ratio of the length of the opposite side to the adjacent side, or $\tan(x) = \frac{\sin(x)}{\cos(x)}$.

On the unit circle, where the radius is 1:

• Sine: The y-coordinate of a point on the unit circle.
• Cosine: The x-coordinate of a point on the unit circle.
• Tangent: The slope of the line connecting the origin to the point on the unit circle.

2. Graphical Representation

The graphs of sine, cosine, and tangent functions exhibit distinct periodic behaviors:

• Sine Function ($y = \sin(x)$): A smooth, continuous wave oscillating between -1 and 1 with a period of $2\pi$.
• Cosine Function ($y = \cos(x)$): Similar to sine but phase-shifted by $\frac{\pi}{2}$ radians.
• Tangent Function ($y = \tan(x)$): Exhibits vertical asymptotes where $\cos(x) = 0$, with a period of $\pi$.

**Amplitude, Period, and Phase Shift:**

• Amplitude: The height from the center line to the peak; for $y = A\sin(x)$ or $y = A\cos(x)$, it is $|A|$.
• Period: The length of one complete cycle; for sine and cosine, it's $2\pi$, and for tangent, $\pi$.
• Phase Shift: Horizontal shift of the graph; altering the argument inside the function shifts the graph left or right.

3. Fundamental Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all valid values of the variable.

**Pythagorean Identity:** $$ \sin^2(x) + \cos^2(x) = 1 $$

**Sum and Difference Formulas:** \begin{align} \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)} \end{align}

**Double Angle Formulas:** \begin{align} \sin(2x) &= 2\sin(x)\cos(x) \\ \cos(2x) &= \cos^2(x) - \sin^2(x) \\ \tan(2x) &= \frac{2\tan(x)}{1 - \tan^2(x)} \end{align}

**Reciprocal Identities:** \begin{align} \csc(x) &= \frac{1}{\sin(x)} \\ \sec(x) &= \frac{1}{\cos(x)} \\ \cot(x) &= \frac{1}{\tan(x)} \end{align}

4. Solving Trigonometric Equations

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

• Using Identities: Simplify equations using trigonometric identities.
• Algebraic Manipulation: Combine like terms and isolate trigonometric functions.
• Graphical Methods: Analyze the graphs of functions to find intersection points.

**Example:** Solve $\sin(x) = \frac{\sqrt{3}}{2}$ for $0 \leq x < 2\pi$.

**Solution:** The solutions are $x = \frac{\pi}{3}$ and $x = \frac{2\pi}{3}$.

5. Applications of Trigonometric Functions

Trigonometric functions model various real-world phenomena:

• Wave Motion: Representing sound and light waves.
• Circular and Oscillatory Motion: Describing the motion of pendulums and springs.
• Engineering and Physics: Analyzing forces, vibrations, and electrical circuits.
• Architecture and Design: Calculating structural loads and aesthetic patterns.

6. Inverse Trigonometric Functions

Inverse trigonometric functions allow for the determination of angles when the values of trigonometric functions are known.

• Arcsine ($\sin^{-1}(x)$): Returns the angle whose sine is $x$.
• Arccosine ($\cos^{-1}(x)$): Returns the angle whose cosine is $x$.
• Arctangent ($\tan^{-1}(x)$): Returns the angle whose tangent is $x$.

These functions are essential in solving triangles and in calculus for integrating and differentiating trigonometric expressions.

7. Trigonometric Form of Complex Numbers

Complex numbers can be expressed in trigonometric (polar) form, which is particularly useful for multiplication, division, and raising to powers.

A complex number $z = a + bi$ can be written as: $$ z = r(\cos(\theta) + i\sin(\theta)) $$ where:

• Magnitude ($r$): $r = \sqrt{a^2 + b^2}$
• Argument ($\theta$): $\theta = \tan^{-1}\left(\frac{b}{a}\right)$

This form leverages Euler's formula: $$ e^{i\theta} = \cos(\theta) + i\sin(\theta) $$

8. Trigonometric Limits and Derivatives

In calculus, trigonometric functions play a crucial role in defining limits and derivatives.

**Limits:** $$ \lim_{x \to 0} \frac{\sin(x)}{x} = 1 $$

**Derivatives:** \begin{align} \frac{d}{dx}[\sin(x)] &= \cos(x) \\ \frac{d}{dx}[\cos(x)] &= -\sin(x) \\ \frac{d}{dx}[\tan(x)] &= \sec^2(x) \end{align}

These foundational derivatives are essential for solving optimization problems and understanding the behavior of functions.

9. Trigonometric Integrals

Integration of trigonometric functions is a key aspect of calculus, involving techniques such as substitution and integration by parts.

**Examples:**

1. $$\int \sin(x) \, dx = -\cos(x) + C$$
2. $$\int \cos(x) \, dx = \sin(x) + C$$
3. $$\int \tan(x) \, dx = -\ln|\cos(x)| + C$$

10. Trigonometric Series and Fourier Analysis

Trigonometric series decompose periodic functions into sums of sines and cosines, a foundation of Fourier analysis.

**Fourier Series:** $$ f(x) = a_0 + \sum_{n=1}^{\infty} [a_n \cos(nx) + b_n \sin(nx)] $$

This decomposition is instrumental in fields such as signal processing, acoustics, and electrical engineering.

Comparison Table

Summary and Key Takeaways