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Overview
Calculus
Computing Systems
Data
Graphs
Linear Algebra
Models and Modeling
Organization
Probability
Programming Constructs
- Programming Constructs Overview
- Abstraction
- Array
- Attribute
- Best-first Search
- Binary Search
- Block
- Branch
- Callback
- Class
- Conditional
- Constructor
- Container/Collection
- Dynamic Array
- Encapsulation
- Exception
- Expression
- Function
- Garbage Collection
- Greedy Algorithms
- Hash
- HTTP Request
- Identifier
- Inheritance
- Inner Class
- Instance
- Iterator
- Keyword
- Lambda
- Libraries
- List
- Linked List
- Literal
- Metaclass
- Method
- Mixin
- Object
- Operator
- Overloading
- Overriding
- Package
- Parameter
- Polymorphism
- Primitive
- Programming Process
- Recursion
- Reflection
- Regular Expression
- Reserved Word
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- Sort
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Statistics
Trigonometry

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Trigonometric Functions

Trigonometric functions relate an angle of a right-angled triangle to ratios of two side lengths.

In the diagram below:

commonly used functions are shown with their associated vectors
the main right-angled triangle is indicated in bold vectors
the hypothenuse h is indicated as length 1

Function Equations

Commonly used trigonometric function equations are:

$\\ sin\left ( \theta \right ) = \frac{y}{h} \\\\cos\left ( \theta \right ) = \frac{x}{h} \\\\tan\left ( \theta \right ) = \frac{y}{x} \\\\cot\left ( \theta \right ) = \frac{x}{y} \\\\sec\left ( \theta \right ) = \frac{h}{x} \\\\csc\left ( \theta \right ) = \frac{h}{y} \\\\arc\left ( \theta \right ) = \frac{\theta }{360}\left ( 2\pi r \right )$

$\\\\\mathrm{\textup{where:}} \\\\sin\,\,\,\,\mathrm{\textup{sine function}} \\cos\,\,\,\,\mathrm{\textup{cosine function}} \\tan\,\,\,\,\mathrm{\textup{tangent function}} \\cot\,\,\,\,\,\mathrm{\textup{cotangent function}} \\sec\,\,\,\,\mathrm{\textup{secant function}} \\csc\,\,\,\,\mathrm{\textup{cosecant function}} \\h\,\,\,\,\,\,\,\,\mathrm{\textup{hypotenuse (h) vector of the right triangle}} \\x\,\,\,\,\,\,\,\,\mathrm{\textup{adjacent (x) vector of the right triangle}} \\y\,\,\,\,\,\,\,\,\mathrm{\textup{opposite (y) vector of the right triangle}} \\r\,\,\,\,\,\,\,\,\mathrm{\textup{radius of the circle, which is also the vector h}} \\\theta \,\,\,\,\,\,\,\,\mathrm{\textup{angle between the h and x vectors of the right triangle}}$

An Example

The above equations can be used to calculate trigonometric values:

$\\\\\mathrm{\textup{assuming:}} \\\\theta =60^{\circ} \\h =1 \\\\\mathrm{\textup{and using a calculator to determine:}} \\\\sin\left ( 60^{\circ} \right ) = .866 \\cos\left ( 60^{\circ} \right ) = .500 \\\\\mathrm{\textup{trigonometric values are:}} \\\\ sin\left ( 60^{\circ} \right ) = \frac{y}{1}=y=.866 \\\\cos\left ( 60^{\circ} \right ) = \frac{x}{1}=x=.500 \\\\tan\left ( 60^{\circ} \right ) = \frac{y}{x}=\frac{.866}{.500}=1.732 \\\\cot\left ( 60^{\circ} \right ) = \frac{x}{y}=\frac{.500}{.866}=.577 \\\\sec\left ( 60^{\circ} \right ) = \frac{h}{x}=\frac{1}{.500}=2.000 \\\\csc\left ( 60^{\circ} \right ) = \frac{h}{y}=\frac{1}{.866}=1.155 \\\\arc\left ( 60^{\circ} \right ) = \frac{60 }{360}\left ( 2\pi 1 \right )=1.047$

References

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