Geometry - Quadrant

• Introduction
• What is Quadrant?
• Four Quadrant Graph
• Sign Convention in Four Quadrants
• Understanding Sign Convention With the Help of Examples
• What is the Meaning of Origin?
• Trigonometric Values in Four Quadrants
• Solved Examples on Quadrant
• Practice Problems
• Frequently Asked Questions

Introduction

Coordinate geometry is a branch of mathematics that enables us to graph the past and present of a phenomenon into simple figures and derive insights for predicting future occurrences. The primary goal of this article is to help readers comprehend the coordinate plane and its quadrants, which are the fundamental elements of coordinate geometry.

The `x`-axis and `y`-axis of the coordinate system constitute a quadrant. The quadrants are formed when the `x`-axis and the `y`-axis connect at a `90^\circ` angle. Coordinates, or negative and positive `x`-axis and `y`-axis values, are included in these regions.

What is Quadrant?

A cartesian plane system with two dimensions divides the plane into `4` indefinite areas known as quadrants using its `x`-axis and `y`-axis. The right angle is generated by the intersection of the `y`-axis, which represents the vertical line, and the `x`-axis, which represents the horizontal line. 

The reference point is frequently used where two lines intersect. This point serves as the reference (or initial point) for all measurements made using the coordinate system. This reference point is called the origin. A quadrant is the cartesian plane region generated when the `y`-axis and the `x`-axis meet.

Four Quadrant Graph

The `x` and `y`-axes intersect at the origin and form `4` quadrants as shown in the following graph.

It should be noted that the quadrants are numbered in an anti-clockwise manner. The place where the `x` and `y`-axes intersect is known as the origin. At the origin, `x` and `y` have the values `(0, 0)`.

Sign Convention in Four Quadrants

Observing the `XY`-plane, the `y`-axis grows when we move from down to up. Similarly, the `x`-axis grows as we move from left to right.

The points on each quadrant of the plane will have different `x` and `y` signs since the plane is split into `4` quadrants. Let's examine the sign conventions in more detail.

For a better understanding, see the quadrant graph below -

Understanding Sign Convention with the Help of Examples

The graph is split into `4` quadrants which are given below:
 

The above coordinate plane shows all the four quadrants.

What Is the Meaning of Origin?

As the starting point for all other points in the plane, the origin is an important concept in the cartesian coordinate system. The intersection of the `x`-axis and `y`-axis, which essentially divides the area into `4` quadrants, is known as the origin. The origin's coordinates are `(0, 0)`, and it serves as the starting point for calculating distances and other points' positions in the coordinate plane.

Numerous mathematical operations and concepts, like the slope formula, midpoint formula, and distance formula, depend critically on the origin. You'll be better able to use the cartesian coordinate system's concepts in everyday life and solve problems involving it if you comprehend the relevance of the origin.

Trigonometric Values in Four Quadrants

Calculus and geometry are the two fields of mathematics where trigonometry is important. Trigonometric angles can be positioned in any of the four quadrants, and depending on the angle's position in the quadrant, the trigonometric functions (sine, cosine, and tangent) have different signs. 

The below table can be used to summarize the trigonometric function signs in each quadrant:

The term  “ASIC” is used to memorize the signs of trigonometric functions in the particular quadrants as already shown in the above diagram.

A - All the trigonometric values are positive in the first quadrant.

S - Only sine and cosecant functions are positive in the second quadrant.

T - Only tangent and cotangent functions are positive in the third quadrant.

C - Only cosine and secant functions are positive in the fourth quadrant.

Solved Examples on Quadrant

Example `1`: Determine the quadrants in which the following coordinates are located.

1. `(4, 13)`
2. `(16, -20)`
3. `(-13, -9)`
4. `(-6,25)`

Solution:

1. Both the `y`-value and `x`-value are positive, so it lies in the Quadrant I.
    
2. In `(16,-20)`, the `y`-value is negative and the `x`-value is positive, So it is located in Quadrant IV.
    
3. Both the `y`-value and `x`-value are negative, so it lies in the Quadrant III.
    
4. Because the `y`-value is positive and the `x`-value is negative, so it lies in the Quadrant II.

Example `2`: Discuss an example of a point that is located in Quadrant I and Quadrant III.

Solution: The coordinates in the `y`-axis and `x`-axis in Quadrant I have positive values. An example of a point in this quadrant is `(+5, +9)`. Similarly, the coordinates in the `y`-axis and `x`-axis in Quadrant III have negative values. An example of a point in this quadrant is `(-5, -9)`.

Example `3`: What quadrant does the origin lie in?

Solution: Since both the `x`- and `y`-axes meet at the origin, represented by the integers `(0, 0)`, which is non-negative, the origin is referred to as being in Quadrant I.

Practice Problems

Q`1`. In a cartesian plane system, how many quadrants are there?

1.  `6`
2.  `4`
3.  `5`
4.  `2`

Answer: b

Q`2`. The coordinate `(-3, -1)` lies in

1. Quadrant II
2. Quadrant IV
3. Quadrant I
4. Quadrant III

Answer: d

Q`3`. Which of the following points is in the fourth quadrant?

1. `(16, -4)`
2. `(-7, 9)`
3. `(10, - 6)` 
4. `(-13, -8)`

Answer: c

Q`4`. For the `y`-axis and the `x`-axis in the second quadrant -

1. Both the axes are positive
2. Both the axes are negative
3. The `y`-axis is negative and the `x`-axis is positive
4. The `y`-axis is positive and the `x`-axis is negative

Answer: d

Frequently Asked Questions

Q`1`. At what point do the four quadrants intersect?

Answer: The origin is the place where the `y`-axis and `x`-axis meet to form `4` quadrants. `(0, 0)` indicates the origin.

Q`2`. What is the first quadrant?

Answer: In a two-dimensional cartesian coordinate system, two axes are split perpendicularly at their intersections to create four equal regions. The first quadrant is the region on the right-top side.

Q`3`. What do you mean by quadrant?

Answer: A quadrant is the area of a cartesian plane where the `y`-axis and `x`-axis converge.