trigonometry

What is Trigonometry?

• Trigonometry is a significant branch of mathematics.

• It is heavily used across various fields.

• The term 'Trigonometry' comes from the Greek words 'trigonon' (triangle) and 'metron' (measure).

• It primarily studies the relationships between the sides and angles of right-angled triangles.

• This discipline aids in finding missing or unknown angles and sides using trigonometric formulas, functions, or identities.

• Angles in trigonometry can be measured in two systems: degrees and radians.

• Commonly used trigonometric angles include:
    - 0°
    - 30°
    - 45°
    - 60°
    - 90°

Sub-Branches of Trigonometry

• Trigonometry is classified into two sub-branches:
    - Plane Trigonometry: Involves two-dimensional figures.
    - Spherical Trigonometry: Involves spherical surfaces and is useful in navigation and astronomy.

Trigonometric Ratios - Sine, Cosine, Tangent

• Trigonometric ratios or functions are pivotal in solving triangles: these include sine (sin), cosine (cos), and tangent (tan).

• In a right-angled triangle:
    - The longest side is the hypotenuse.
    - The side opposite the angle of interest is the opposite side.
    - The remaining side that forms the angle is the adjacent side.

Six Important Trigonometric Functions

• The six critical trigonometric functions can be defined in the context of a right-angled triangle:
    - Sine Function (sin):
      - Relation: $ext{sin}( heta) = \frac{ ext{Opposite Side}}{ ext{Hypotenuse}}$
    - Cosine Function (cos):
      - Relation: $ext{cos}( heta) = \frac{ ext{Adjacent Side}}{ ext{Hypotenuse}}$
    - Tangent Function (tan):
      - Relation: $ext{tan}( heta) = \frac{ ext{Opposite Side}}{ ext{Adjacent Side}}$
    - Cosecant Function (cosec):
      - Relation: $ext{cosec}( heta) = \frac{ ext{Hypotenuse}}{ ext{Opposite Side}}$
    - Secant Function (sec):
      - Relation: $ext{sec}( heta) = \frac{ ext{Hypotenuse}}{ ext{Adjacent Side}}$
    - Cotangent Function (cot):
      - Relation: $ext{cot}( heta) = \frac{ ext{Adjacent Side}}{ ext{Opposite Side}}$

Even and Odd Trigonometric Functions

• Trigonometric functions can be categorized as either even or odd:
    - Odd Functions: A function $f$ is odd if:
      - $f(-x) = -f(x)$
      - Symmetric with respect to the origin.
    - Even Functions: A function $f$ is even if:
      - $f(-x) = f(x)$
      - Symmetric with respect to the y-axis.
    - Examples include:
      - $ext{sin}(-x) = - ext{sin}(x)$ (Odd)
      - $ext{cos}(-x) = ext{cos}(x)$ (Even)
      - $ext{tan}(-x) = - ext{tan}(x)$ (Odd)
      - $ext{csc}(-x) = - ext{csc}(x)$ (Odd)
      - $ext{sec}(-x) = ext{sec}(x)$ (Even)
      - $ext{cot}(-x) = - ext{cot}(x)$ (Odd)
    - Understanding even and odd functions simplifies the evaluation of expressions.

Important Trigonometric Angles

• Commonly utilized angles in trigonometry are:
    - 0°, 30°, 45°, 60°, 90°.

• Each angle corresponds to specific well-known trigonometric ratios, often taught through memorization techniques.

• For example, in a right-angle triangle:
    - Sine (), Cosine (
  ), and Tangent can be calculated as:
      - $ext{sin}( heta) = \frac{ ext{Perpendicular}}{ ext{Hypotenuse}}$
      - $heta = ext{sin}^{-1}\left(\frac{P}{H}\right)$
      - $heta = ext{cos}^{-1}\left(\frac{B}{H}\right)$
      - $heta = ext{tan}^{-1}\left(\frac{P}{B}\right)$

Trigonometry Table

• A table for common angles' trigonometric ratios:
    - | Angle | 0° | 30° | 45° | 60° | 90° |
    - |---------|----------|---------------|----------|------------|-------|
    - | Sin | 0 | $\frac{1}{2}$ | $\frac{1}{\sqrt{2}}$ | $\frac{\sqrt{3}}{2}$ | 1 |
    - | Cos | 1 | $\frac{\sqrt{3}}{2}$ | $\frac{1}{\sqrt{2}}$ | $\frac{1}{2}$ | 0 |
    - | Tan | 0 | $\frac{1}{\sqrt{3}}$ | 1 | $\sqrt{3}$ | Undefined |
    - | Cosec | Undefined| 2 | $\sqrt{2}$ | $\frac{2}{\sqrt{3}}$ | 1 |
    - | Sec | 1 | $\frac{2}{\sqrt{3}}$ | $\sqrt{2}$ | 2 | Undefined |
    - | Cot | Undefined| $\sqrt{3}$ | 1 | $\frac{1}{\sqrt{3}}$ | 0 |
    - The ratios can also extend beyond 90 degrees such as:
      - 180°, 270°, and 360°.

Unit Circle

• The Unit Circle provides a graphical representation for trigonometric ratios:
    - A circle with radius 1 centered at the origin (0,0).
    - For angle $\theta$, coordinates on the unit circle represent:
      - $(\cos(\theta), \sin(\theta))$
      - Relationship of lengths:
        - $\text{Sin} \theta = \frac{y}{1} = y$
        - $\text{Cos} \theta = \frac{x}{1} = x$
        - $\text{Tan} \theta = \frac{y}{x}$

List of Trigonometric Formulas

• Trigonometric identities are equations that hold true for right-angled triangles and include:
    - Pythagorean Identities:
      - $\sin^2(\theta) + \cos^2(\theta) = 1$
      - $\tan^2(\theta) + 1 = \sec^2(\theta)$
      - $\cot^2(\theta) + 1 = \csc^2(\theta)$
      - Double angle identities:
        - $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$
        - $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$
        - $\tan(2\theta) = \frac{2\tan(\theta)}{1-\tan^2(\theta)}$
    - Sum and Difference Identities (for angles u and v):
      - $\sin(u + v) = \sin(u)\cos(v) + \cos(u)\sin(v)$
      - $\cos(u + v) = \cos(u)\cos(v) - \sin(u)\sin(v)$
      - $\tan(u + v) = \frac{\tan(u) + \tan(v)}{1 - \tan(u)\tan(v)}$
    - Sine and Cosine Laws:
      - If A, B, and C are triangle angles, and a, b, and c are opposite sides:
        - $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
      - Cosine Laws:
        - $c^2 = a^2 + b^2 - 2ab\cos C$
        - $a^2 = b^2 + c^2 - 2bc\cos A$
        - $b^2 = a^2 + c^2 - 2ac\cos B$

Trigonometry Basics

• The three principal trigonometric functions: sine, cosine, and tangent are fundamental for understanding trigonometrical concepts.

• The derived functions (cotangent, secant, cosecant) relate as follows:
    - $\text{cot} \theta = \frac{1}{\text{tan} \theta}$
    - $\text{sec} \theta = \frac{1}{\text{cos} \theta}$
    - $\text{cosec} \theta = \frac{1}{\text{sin} \theta}$

• Definitions based on a right-angled triangle:
    - $\text{Sin} \theta = \frac{\text{Perpendicular}}{\text{Hypotenuse}}$
    - $\text{Cos} \theta = \frac{\text{Base}}{\text{Hypotenuse}}$
    - $\text{Tan} \theta = \frac{\text{Perpendicular}}{\text{Base}}$

• Relationships give insight into triangle dimensions regarding their angles and proportions.

Trigonometry Applications

• Trigonometry is widely applied in fields such as:
    - Oceanography
    - Seismology
    - Meteorology
    - Physical sciences
    - Astronomy
    - Acoustics
    - Navigation
    - Electronics

• Practical use cases include determining the height of objects like buildings and mountains using angles.

• For example, using the tangent function to determine the distance from a building, applying:
    - $\tan(\alpha) = \frac{\text{Height}}{\text{Distance}}$
    - Given height = 20 m and angle = 45 degrees:
      - $\text{Distance} = \frac{\text{Height}}{\tan(45^{\circ})}$
      - Since $\tan(45^{\circ}) = 1$, therefore, Distance = 20 m.