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Coordinate Systems

- Calculus
- Absolute Maxima and Minima
- Accumulation Function
- Accumulation Problems
- Algebraic Functions
- Alternating Series
- Application of Derivatives
- Approximating Areas
- Arc Length of a Curve
- Arithmetic Series
- Average Value of a Function
- Calculus of Parametric Curves
- Candidate Test
- Combining Differentiation Rules
- Continuity
- Continuity Over an Interval
- Convergence Tests
- Cost and Revenue
- Derivative Functions
- Derivative of Exponential Function
- Derivative of Inverse Function
- Derivative of Logarithmic Functions
- Derivative of Trigonometric Functions
- Derivatives
- Derivatives and Continuity
- Derivatives and the Shape of a Graph
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Polar Functions
- Derivatives of Sin, Cos and Tan
- Determining Volumes by Slicing
- Disk Method
- Divergence Test
- Euler's Method
- Evaluating a Definite Integral
- Evaluation Theorem
- Exponential Functions
- Finding Limits
- Finding Limits of Specific Functions
- First Derivative Test
- Function Transformations
- Geometric Series
- Growth Rate of Functions
- Higher-Order Derivatives
- Hyperbolic Functions
- Implicit Differentiation Tangent Line
- Improper Integrals
- Initial Value Problem Differential Equations
- Integral Test
- Integrals of Exponential Functions
- Integrals of Motion
- Integrating Even and Odd Functions
- Integration Tables
- Integration Using Long Division
- Integration of Logarithmic Functions
- Integration using Inverse Trigonometric Functions
- Intermediate Value Theorem
- Inverse Trigonometric Functions
- Jump Discontinuity
- Limit Laws
- Limit of Vector Valued Function
- Limit of a Sequence
- Limits
- Limits at Infinity
- Limits of a Function
- Linear Differential Equation
- Logarithmic Differentiation
- Logarithmic Functions
- Logistic Differential Equation
- Maclaurin Series
- Maxima and Minima
- Maxima and Minima Problems
- Mean Value Theorem for Integrals
- Models for Population Growth
- Motion Along a Line
- Natural Logarithmic Function
- Net Change Theorem
- Newton's Method
- One-Sided Limits
- Optimization Problems
- P Series
- Particular Solutions to Differential Equations
- Polar Coordinates Functions
- Polar Curves
- Population Change
- Power Series
- Ratio Test
- Removable Discontinuity
- Riemann Sum
- Rolle's Theorem
- Root Test
- Second Derivative Test
- Separable Equations
- Simpson's Rule
- Solid of Revolution
- Solutions to Differential Equations
- Surface Area of Revolution
- Tangent Lines
- Taylor Series
- Techniques of Integration
- The Fundamental Theorem of Calculus
- The Mean Value Theorem
- The Power Rule
- The Squeeze Theorem
- The Trapezoidal Rule
- Theorems of Continuity
- Trigonometric Substitution
- Vector Valued Function
- Vectors in Calculus
- Washer Method
- Decision Maths
- Geometry
- 2 Dimensional Figures
- 3 Dimensional Vectors
- 3-Dimensional Figures
- Altitude
- Angles in Circles
- Arc Measures
- Area and Volume
- Area of Circles
- Area of Circular Sector
- Area of Parallelograms
- Area of Plane Figures
- Area of Rectangles
- Area of Regular Polygons
- Area of Rhombus
- Area of Trapezoid
- Area of a Kite
- Composition
- Congruence Transformations
- Congruent Triangles
- Convexity in Polygons
- Coordinate Systems
- Dilations
- Distance and Midpoints
- Equation of Circles
- Equilateral Triangles
- Figures
- Fundamentals of Geometry
- Geometric Inequalities
- Geometric Mean
- Geometric Probability
- Glide Reflections
- HL ASA and AAS
- Identity Map
- Inscribed Angles
- Isometry
- Isosceles Triangles
- Law of Cosines
- Law of Sines
- Linear Measure and Precision
- Median
- Parallel Lines Theorem
- Parallelograms
- Perpendicular Bisector
- Plane Geometry
- Polygons
- Projections
- Properties of Chords
- Proportionality Theorems
- Pythagoras Theorem
- Rectangle
- Reflection in Geometry
- Regular Polygon
- Rhombuses
- Right Triangles
- Rotations
- SSS and SAS
- Segment Length
- Similarity
- Similarity Transformations
- Special quadrilaterals
- Squares
- Surface Area of Cone
- Surface Area of Cylinder
- Surface Area of Prism
- Surface Area of Sphere
- Surface Area of a Solid
- Surface of Pyramids
- Symmetry
- Translations
- Trapezoids
- Triangle Inequalities
- Triangles
- Using Similar Polygons
- Vector Addition
- Vector Product
- Volume of Cone
- Volume of Cylinder
- Volume of Pyramid
- Volume of Solid
- Volume of Sphere
- Volume of prisms
- Mechanics Maths
- Acceleration and Time
- Acceleration and Velocity
- Assumptions
- Calculus Kinematics
- Coefficient of Friction
- Connected Particles
- Constant Acceleration
- Constant Acceleration Equations
- Converting Units
- Force as a Vector
- Kinematics
- Newton's First Law
- Newton's Second Law
- Newton's Third Law
- Projectiles
- Pulleys
- Resolving Forces
- Statics and Dynamics
- Tension in Strings
- Variable Acceleration
- Probability and Statistics
- Bar Graphs
- Basic Probability
- Charts and Diagrams
- Conditional Probabilities
- Continuous and Discrete Data
- Frequency, Frequency Tables and Levels of Measurement
- Independent Events Probability
- Line Graphs
- Mean Median and Mode
- Mutually Exclusive Probabilities
- Probability Rules
- Probability of Combined Events
- Quartiles and Interquartile Range
- Systematic Listing
- Pure Maths
- ASA Theorem
- Absolute Value Equations and Inequalities
- Addition and Subtraction of Rational Expressions
- Addition, Subtraction, Multiplication and Division
- Algebra
- Algebraic Fractions
- Algebraic Notation
- Algebraic Representation
- Analyzing Graphs of Polynomials
- Angle Measure
- Angles
- Angles in Polygons
- Approximation and Estimation
- Area and Circumference of a Circle
- Area and Perimeter of Quadrilaterals
- Area of Triangles
- Arithmetic Sequences
- Average Rate of Change
- Bijective Functions
- Binomial Expansion
- Binomial Theorem
- Chain Rule
- Circle Theorems
- Circles
- Circles Maths
- Combination of Functions
- Common Factors
- Common Multiples
- Completing the Square
- Completing the Squares
- Complex Numbers
- Composite Functions
- Composition of Functions
- Compound Interest
- Compound Units
- Construction and Loci
- Converting Metrics
- Convexity and Concavity
- Coordinate Geometry
- Coordinates in Four Quadrants
- Cubic Function Graph
- Cubic Polynomial Graphs
- Data transformations
- Deductive Reasoning
- Definite Integrals
- Deriving Equations
- Determinant of Inverse Matrix
- Determinants
- Differential Equations
- Differentiation
- Differentiation Rules
- Differentiation from First Principles
- Differentiation of Hyperbolic Functions
- Direct and Inverse proportions
- Disjoint and Overlapping Events
- Disproof by Counterexample
- Distance from a Point to a Line
- Divisibility Tests
- Double Angle and Half Angle Formulas
- Drawing Conclusions from Examples
- Ellipse
- Equation of Line in 3D
- Equation of a Perpendicular Bisector
- Equation of a circle
- Equations
- Equations and Identities
- Equations and Inequalities
- Estimation in Real Life
- Euclidean Algorithm
- Evaluating and Graphing Polynomials
- Even Functions
- Exponential Form of Complex Numbers
- Exponential Rules
- Exponentials and Logarithms
- Expression Math
- Expressions and Formulas
- Faces Edges and Vertices
- Factorials
- Factoring Polynomials
- Factoring Quadratic Equations
- Factorising expressions
- Factors
- Finding Maxima and Minima Using Derivatives
- Finding Rational Zeros
- Finding the Area
- Forms of Quadratic Functions
- Fractional Powers
- Fractional Ratio
- Fractions
- Fractions and Decimals
- Fractions and Factors
- Fractions in Expressions and Equations
- Fractions, Decimals and Percentages
- Function Basics
- Functional Analysis
- Functions
- Fundamental Counting Principle
- Fundamental Theorem of Algebra
- Generating Terms of a Sequence
- Geometric Sequence
- Gradient and Intercept
- Graphical Representation
- Graphing Rational Functions
- Graphing Trigonometric Functions
- Graphs
- Graphs and Differentiation
- Graphs of Common Functions
- Graphs of Exponents and Logarithms
- Graphs of Trigonometric Functions
- Greatest Common Divisor
- Growth and Decay
- Growth of Functions
- Highest Common Factor
- Hyperbolas
- Imaginary Unit and Polar Bijection
- Implicit differentiation
- Inductive Reasoning
- Inequalities Maths
- Infinite geometric series
- Injective functions
- Instantaneous Rate of Change
- Integers
- Integrating Polynomials
- Integrating Trig Functions
- Integrating e^x and 1/x
- Integration
- Integration Using Partial Fractions
- Integration by Parts
- Integration by Substitution
- Integration of Hyperbolic Functions
- Interest
- Inverse Hyperbolic Functions
- Inverse and Joint Variation
- Inverse functions
- Iterative Methods
- Law of Cosines in Algebra
- Law of Sines in Algebra
- Laws of Logs
- Limits of Accuracy
- Linear Expressions
- Linear Systems
- Linear Transformations of Matrices
- Location of Roots
- Logarithm Base
- Logic
- Lower and Upper Bounds
- Lowest Common Denominator
- Lowest Common Multiple
- Math formula
- Matrices
- Matrix Addition and Subtraction
- Matrix Determinant
- Matrix Multiplication
- Metric and Imperial Units
- Misleading Graphs
- Mixed Expressions
- Modulus Functions
- Modulus and Phase
- Multiples of Pi
- Multiplication and Division of Fractions
- Multiplicative Relationship
- Multiplying and Dividing Rational Expressions
- Natural Logarithm
- Natural Numbers
- Notation
- Number
- Number Line
- Number Systems
- Numerical Methods
- Odd functions
- Open Sentences and Identities
- Operation with Complex Numbers
- Operations with Decimals
- Operations with Matrices
- Operations with Polynomials
- Order of Operations
- Parabola
- Parallel Lines
- Parametric Differentiation
- Parametric Equations
- Parametric Integration
- Partial Fractions
- Pascal´s Triangle
- Percentage
- Percentage Increase and Decrease
- Percentage as fraction or decimals
- Perimeter of a Triangle
- Permutations and Combinations
- Perpendicular Lines
- Points Lines and Planes
- Polynomial Graphs
- Polynomials
- Powers Roots And Radicals
- Powers and Exponents
- Powers and Roots
- Prime Factorization
- Prime Numbers
- Problem-solving Models and Strategies
- Product Rule
- Proof
- Proof and Mathematical Induction
- Proof by Contradiction
- Proof by Deduction
- Proof by Exhaustion
- Proof by Induction
- Properties of Exponents
- Proportion
- Proving an Identity
- Pythagorean Identities
- Quadratic Equations
- Quadratic Function Graphs
- Quadratic Graphs
- Quadratic functions
- Quadrilaterals
- Quotient Rule
- Radians
- Radical Functions
- Rates of Change
- Ratio
- Ratio Fractions
- Rational Exponents
- Rational Expressions
- Rational Functions
- Rational Numbers and Fractions
- Ratios as Fractions
- Real Numbers
- Reciprocal Graphs
- Recurrence Relation
- Recursion and Special Sequences
- Remainder and Factor Theorems
- Representation of Complex Numbers
- Rewriting Formulas and Equations
- Roots of Complex Numbers
- Roots of Polynomials
- Rounding
- SAS Theorem
- SSS Theorem
- Scale Drawings and Maps
- Scale Factors
- Scientific Notation
- Sector of a Circle
- Segment of a Circle
- Sequences
- Sequences and Series
- Series Maths
- Sets Math
- Similar Triangles
- Similar and Congruent Shapes
- Simple Interest
- Simplifying Fractions
- Simplifying Radicals
- Simultaneous Equations
- Sine and Cosine Rules
- Small Angle Approximation
- Solving Linear Equations
- Solving Linear Systems
- Solving Quadratic Equations
- Solving Radical Inequalities
- Solving Rational Equations
- Solving Simultaneous Equations Using Matrices
- Solving Systems of Inequalities
- Solving Trigonometric Equations
- Solving and Graphing Quadratic Equations
- Solving and Graphing Quadratic Inequalities
- Special Products
- Standard Form
- Standard Integrals
- Standard Unit
- Straight Line Graphs
- Substraction and addition of fractions
- Sum and Difference of Angles Formulas
- Surds
- Surjective functions
- Tables and Graphs
- Tangent of a Circle
- The Quadratic Formula and the Discriminant
- Transformations
- Transformations of Graphs
- Translations of Trigonometric Functions
- Triangle Rules
- Triangle trigonometry
- Trigonometric Functions
- Trigonometric Functions of General Angles
- Trigonometric Identities
- Trigonometric Ratios
- Trigonometry
- Turning Points
- Types of Functions
- Types of Numbers
- Types of Triangles
- Unit Circle
- Units
- Variables in Algebra
- Vectors
- Verifying Trigonometric Identities
- Writing Equations
- Writing Linear Equations
- Statistics
- Binomial Distribution
- Binomial Hypothesis Test
- Bivariate Data
- Box Plots
- Categorical Data
- Categorical Variables
- Central Limit Theorem
- Comparing Data
- Conditional Probability
- Correlation Math
- Cumulative Frequency
- Data Interpretation
- Discrete Random Variable
- Distributions
- Events (Probability)
- Frequency Polygons
- Geometric Distribution
- Histograms
- Hypothesis Test for Correlation
- Hypothesis Testing
- Large Data Set
- Linear Interpolation
- Measures of Central Tendency
- Methods of Data Collection
- Normal Distribution
- Normal Distribution Hypothesis Test
- Probability
- Probability Calculations
- Probability Distribution
- Probability Generating Function
- Quantitative Variables
- Random Variables
- Sampling
- Scatter Graphs
- Single Variable Data
- Standard Deviation
- Standard Normal Distribution
- Statistical Measures
- Tree Diagram
- Type I Error
- Type II Error
- Types of Data in Statistics
- Venn Diagrams

Being able to describe something's geometric position is fundamental to a useful and robust mathematical system. So how exactly have we as humans taken to doing this? Well, we use **coordinate systems** of course! From maps to graphs, and the pixels on the screen you are viewing this on, coordinate systems are everywhere in our everyday lives if you look closely enough, and life just wouldn't be the same without them!

Coordinate systems are systems employed to describe the position of points in a defined space using one or multiple numbers.

There are many types of coordinate systems, some may be very familiar, such as simple number lines, and others you may not be so familiar with, such as polar coordinate systems. The important thing that all coordinate systems have in common, is that they give us a means of describing where a point is in space, relative to some *origin*, i.e. the zero point of the system.

The number line below is a very simple example of a coordinate system.

The position of a point on the number line can be expressed as a single number. This number essentially denotes the horizontal distance of that point from the origin, i.e. where the is marked. The position of any point in the coordinate system is made in reference to the origin.

For instance, point on the number line below is 3 units from the origin.

Now, let's take a closer look at some of the main coordinate systems you might come across.

There are two primary types of coordinate systems in mathematics: the cartesian coordinate system and the polar coordinate system. The cartesian coordinate system is defined by distances along a set of perpendicular number lines, whereas the polar coordinate system is defined by an angle and radial distance. Let's dig a little deeper into each to get a better understanding.

By combining one vertical and one horizontal number line, we get the cartesian coordinate system. Each of these number lines is known as an *axis*, together creating a plane known as the *coordinate plane*.

Any point on the coordinate plane can be described using two numbers, one to describe the distance along the horizontal axis, also known as the *x**-axis*, and the other to describe the distance up the vertical axis, also known as the *y-axis*. Each of these numbers is known as a coordinate.

The notation for the coordinates of a point in the cartesian coordinate system is simply a pair of numbers within brackets, the first representing the distance of a point from the origin along the *x*-axis, and the other representing the distance of a point from the origin along the *y*-axis.

For instance, the point on the below cartesian coordinate system has coordinates , as it is steps along the -axis, and steps up the -axis.

By extending each axis, and to include numbers less than zero, i.e. negative numbers, we can reveal the full scope of the cartesian coordinate plane.

If the point is located in the upper-right quadrant the two coordinates x and y will be positive.

If the point is located in the upper-left quadrant then the x coordinate will be negative and the y coordinate will be positive.

If the point is located in the lower-left quadrant then the two coordinates x and y will be negative.

Finally, if the point is located in the lower-right quadrant then the x coordinate will be positive and the Y coordinate will be negative.

**(1)**

What are the coordinates of point C in the cartesian coordinate system below?

**Solution:**

By noticing that point C is located in the upper-left quadrant, we can assert that it must have a positive coordinate and negative coordinate.

From observation, we can deduce that the *x *coordinate is, as the perpendicular horizontal distance between point C and the y-axis is 3 units.

On the other hand, the *y* coordinate is, since the perpendicular vertical distance between point C and the x-axis is 5 units.

Therefore, the cartesian coordinates of point C are

**(2)**

What are the coordinates of point D in the cartesian coordinate system below?

**Solution:**

By noticing that point D is located in the lower-left quadrant, we can assert that it must have a negative coordinate and a negative coordinate.

From observation, we can deduce that the *x* coordinate is, as the perpendicular horizontal distance between point D and the y-axis isunits.

On the other hand, the *y *coordinate is since the perpendicular vertical distance between point C and the x-axis is units.

Therefore, the cartesian coordinates of point C are

It is worth mentioning that 2 special cases will occur when the point is located on the x or the y-axes.

This concept can be better explained by the following two examples:

**(3)**

What are the coordinates of point E in the cartesian coordinate system below?

**Solution:**

Since the point E actually lies on the *y*-axis, the distance between it and the origin along the *x*-axis is in fact therefore the *x* coordinate is.

On the other hand, since the vertical distance between point E and the origin isunits, it is clear that the *y* coordinate is.

So, we are able to conclude that the cartesian coordinates of E are

**(4)**

What are the coordinates of point F in the cartesian coordinate system below?

**Solution:**

Since the point F lies on the *x*-axis, the distance between it and the origin along the *y*-axis is therefore the *y* coordinate is

On the other hand, since the horizontal distance between point F and the origin is units, it is clear that the *x* coordinate is

So, we are able to conclude that the cartesian coordinates of E are

After the two examples given before, we can conclude the following:

If a point is located on the y-axis, then thecoordinate is

If a point is located on the x-axis, then thecoordinate is

The polar coordinate system bears similarities to the cartesian coordinate system in that the position of any point can be defined by two numbers. However, rather than these two numbers indicating distance along perpendicular axes, in the case of polar coordinates, the two numbers indicate **radial distance** and **angular distance**.

What exactly do we mean by this? Let's take a look at a polar coordinate system to find out!

You can see that the polar coordinate system below, rather than being made up of two perpendicular axes, is in fact made up of many concentric circles, with radial lines outward from their common center indicating angles.

Any point on this coordinate system can be found by first moving along the number line at by the desired amount, and then performing a circular rotation. In essence, the two coordinates are a radius, , and angle . This is written as

Let's take point A below as an example. To reach point A, we simply move along the number line units, and then make a rotation through

We could, therefore, say that the polar coordinates of point A are However, more often coordinates in polar coordinate systems are given in **radians** rather than degrees.

To convert degrees to radians we simply multiply by Therefore, in radians is

So our new polar coordinates for point A are

Let's take a look at another example to make sure we've got it!

What are the polar coordinates of points B and C in the polar coordinate system below? Angles should be converted to radians.

**Solution:**

Taking point B first, we can see that it can be reached by moving along units along the number line at and then rotating an angular distance of Therefore, the polar coordinates of point B are

To convert to radians we simply multiply it by and so

Therefore the polar coordinates of point B are .

Now we simply do the same for point C. We see that it can be reached by moving along units along the number line at and then rotating through an angular distance of Therefore the polar coordinates of point C are .

But wait! a rotation of is the same as a rotation of . Therefore the polar coordinates can also be written as By multiplying each angle by we get that point C has polar coordinates or alternatively

How exactly do we convert between coordinates in cartesian and polar coordinate systems though? Let's take a look!

Converting between the two coordinate systems is simple if we employ some of our knowledge of trigonometry.

Consider the following right-angled triangle in the cartesian coordinate plane. On this triangle, point A's cartesian and polar coordinates have been marked, and respectively.

Using simple trigonometry, we can note the following equations to be true

and

These equations allow us to convert from polar coordinates to cartesian coordinates.

Considering Pythagoras theorem, we can find the following equation for

Finding angular position, , is then a simple case of employing trigonometry once again

or

Let's take a look at a couple of examples to make sure we've got it!

**(1)**

Convert the cartesian coordinates, , into polar coordinates.

**Solution:**

To find the coordinate we simply use Pythagoras' theorem

Now to find the coordinate, we use trigonometry

Therefore the polar coordinates are

If you're struggling to get the correct angular coordinate, remember to set your calculator to radians!

**(2)**

Convert the following polar coordinates, to cartesian coordinates.

**Solution:**

To convert from polar coordinates to cartesian, we use the following simple trigonometric equations.

and

Starting with the coordinate

And then the coordinate

Therefore the cartesian coordinates are

Coordinate systems are frequently used to represent the location of a point, but they can also be used to specify the location of more complex shapes like lines, planes, circles, or spheres.

Perhaps the most important and oldest use of coordinate systems is in maps. World maps use a special geographic coordinate system that bears similarities to the cartesian coordinate system. Positions in world maps are made up of a **latitude **coordinate, and a **longitude** coordinate.

As mentioned earlier in the explanation, screens use coordinate systems as well. Each pixel in a screen has a coordinate that specifies its place horizontally and vertically. This gives each pixel a unique identifier with which it can be located and controlled. This is all going on in the background of the device you are reading this on now!

A modified version of polar coordinates is used in many forms of navigation. Polar coordinates are ideal for navigation as relative position can be defined by the desired angle of movement and the distance between points. The **velocity-heading** model is a guidance system used in missiles intercepting a moving target, and is based on the idea of polar coordinates!

- Coordinate systems are used to locate the position of a point in the plane.
- In two-dimensional coordinate systems, each point is represented by two integers.
- Two common coordinate systems are Cartesian and polar coordinate systems.
- We can convert between polar and cartesian coordinate systems using a mixture of trigonometry and Pythagoras' theorem

Coordinate systems are systems used to define the geometric position of points.

Cartesian and Polar coordinate systems.

Coordinate systems are used in many aspects of everyday life from maps to tv screens.

More about Coordinate Systems

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