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Function

Angle Value

Angle Unit

Degrees Radians Gradians

Quick Angles

Unit Circle

Right Triangle

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Historical Background

Babylonians (1900 BCE): Used tables of chords for astronomical calculations.
Greeks (Hipparchus, Ptolemy): Developed early trigonometric tables.
Indians (Aryabhata, Brahmagupta): Introduced sine and cosine functions.
Islamic mathematicians (Al-Battani, Al-Tusi): Expanded trigonometric identities.
Europe (16th–17th centuries): Trigonometry became central to navigation and calculus.

Fundamental Concepts

Angles

Measured in degrees (°) or radians.
360∘=2π radians360^\circ = 2\pi \, \text{radians}.

Right Triangle Trigonometry

For a right triangle with angle θ\theta:

sin⁡θ=oppositehypotenuse,cos⁡θ=adjacenthypotenuse,tan⁡θ=oppositeadjacent\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan \theta = \frac{\text{opposite}}{\text{adjacent}}

Mnemonic: SOH-CAH-TOA.

The Six Trigonometric Functions

Function	Definition	Reciprocal
Sine (sin⁡\sin)	Opposite / Hypotenuse	Cosecant (csc⁡\csc)
Cosine (cos⁡\cos)	Adjacent / Hypotenuse	Secant (sec⁡\sec)
Tangent (tan⁡\tan)	Opposite / Adjacent	Cotangent (cot⁡\cot)

The Unit Circle

Circle of radius 1 centered at the origin.
Coordinates (x,y)(x, y) correspond to (cos⁡θ,sin⁡θ)(\cos \theta, \sin \theta).
Defines trig functions for all real numbers.

Trigonometric Identities

Pythagorean Identities

sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1

1+tan⁡2θ=sec⁡2θ1 + \tan^2 \theta = \sec^2 \theta

1+cot⁡2θ=csc⁡2θ1 + \cot^2 \theta = \csc^2 \theta

Angle Sum and Difference

sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡B\sin (A \pm B) = \sin A \cos B \pm \cos A \sin B

cos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡B\cos (A \pm B) = \cos A \cos B \mp \sin A \sin B

Double Angle

sin⁡2θ=2sin⁡θcos⁡θ\sin 2\theta = 2\sin \theta \cos \theta

cos⁡2θ=cos⁡2θ−sin⁡2θ\cos 2\theta = \cos^2 \theta – \sin^2 \theta

Laws of Sines and Cosines

Law of Sines

asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

Law of Cosines

c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 – 2ab \cos C

Graphs of Trigonometric Functions

Sine and Cosine: Periodic with period 2π2\pi.
Tangent and Cotangent: Periodic with period π\pi.
Secant and Cosecant: Reciprocal graphs with asymptotes.

Inverse Trigonometric Functions

sin⁡−1x,cos⁡−1x,tan⁡−1x\sin^{-1} x, \cos^{-1} x, \tan^{-1} x.
Restricted domains to ensure one-to-one mapping.
Used to solve equations and find angles.

Advanced Topics

Euler’s Formula:

eiθ=cos⁡θ+isin⁡θe^{i\theta} = \cos \theta + i \sin \theta

Complex Numbers: Trigonometric form of complex numbers.
Hyperbolic Functions: sinh⁡,cosh⁡,tanh⁡\sinh, \cosh, \tanh.
Fourier Analysis: Representing signals as sums of sines and cosines.

Applications of Trigonometry

Astronomy: Calculating distances to stars and planets.
Navigation: Determining position using triangulation.
Engineering: Structural analysis, wave mechanics.
Physics: Oscillations, optics, electromagnetism.
Computer Graphics: Rotations, 3D modeling, animation.
Agriculture: Land surveying, irrigation planning.

Worked Examples

Example 1: Height of a Building

If the angle of elevation is 30∘30^\circ and distance from building is 50m:

tan⁡30∘=h50⇒h=50⋅tan⁡30∘=28.9 m\tan 30^\circ = \frac{h}{50} \quad \Rightarrow \quad h = 50 \cdot \tan 30^\circ = 28.9 \, m

Example 2: Solving a Triangle

Given a=8,b=6,C=60∘a=8, b=6, C=60^\circ:

c2=82+62−2(8)(6)cos⁡60∘=64+36−48=52c^2 = 8^2 + 6^2 – 2(8)(6)\cos 60^\circ = 64 + 36 – 48 = 52

c=52≈7.21c = \sqrt{52} \approx 7.21

Comparative Table: Trigonometric vs Hyperbolic Functions

Function	Trigonometric	Hyperbolic
Sine	sin⁡x=eix−e−ix2i\sin x = \frac{e^{ix} – e^{-ix}}{2i}	sinh⁡x=ex−e−x2\sinh x = \frac{e^x – e^{-x}}{2}
Cosine	cos⁡x=eix+e−ix2\cos x = \frac{e^{ix} + e^{-ix}}{2}	cosh⁡x=ex+e−x2\cosh x = \frac{e^x + e^{-x}}{2}
Tangent	tan⁡x=sin⁡xcos⁡x\tan x = \frac{\sin x}{\cos x}	tanh⁡x=sinh⁡xcosh⁡x\tanh x = \frac{\sinh x}{\cosh x}

FAQs

Q: What is trigonometry used for in real life? A: Navigation, engineering, physics, architecture, computer graphics, and more.

Q: What are the three main trigonometric ratios? A: Sine, cosine, and tangent.

Q: Why is the unit circle important? A: It defines trig functions for all real numbers and simplifies identities.

Q: What is Euler’s formula? A: eiθ=cos⁡θ+isin⁡θe^{i\theta} = \cos \theta + i \sin \theta, linking exponential and trigonometric functions.

Q: How is trigonometry used in agriculture? A: For land surveying, irrigation design, and optimizing crop layouts.