In this post we discuss the signs of trigonometry function in various coordinates. Every function in trigonometry has variable character, its polarity (+ve/-ve) changes on the basis of the position of coordinates. In the below section we have tried to explain this concept such that even the student with weak math background will understand.

Not that the concept is significant for Grade 11 students as it will set the base to solve advanced questions asked in Grade 12 or in your professional courses, so make sure you invest some tie in this chapter.

**Quadrants**

**What are Quadrants**

In a Cartesian plane there are two axis (axis x and axis y) which jointly make up 4 sections, each section in the plane is known as Quadrant.

Let us understand the concept with the help of illustration

In the above illustration you can see two axis (x axis and y axis), both the axis meet at a common point O.

X axis at the right of point O shows** positive numbe**r

X axis on the left side of point O shows **negative number**

Y axis above the point O shows **positive number**Y axis below point O shows

**negative numbe**r

Both the axis form 4 different quadrant in the plane.

You can see in the diagram above that there are 4 quadrant in x-y plane**1st Quadrant **–> **Both x and y points are positive****2nd Quadrant** –> ** x is negative and y is positive****3rd Quadrant** –> **Both x and y are negative****4th Quadrant** –> **x is positive and y is negative**

**Example 01**

In the illustration above check point A (2 , 1) in first quadrant

you can clearly see that value of x is 2 and value of y is 1, both the coordinates are positive

**Example 2**

Check point B (-1, 2) in second quadrant

value of x is -1 and value of y is 2, hence x coordinate is negative and y coordinate is positive

I hope i am clear with the concept till now.

Next we will understand the signs/polarity of trigonometry function in different coordinates.

**Signs of Trigonometry Function**

In the below illustration, I have shown the sign of trigonometry function in each coordinate plane

From the above figure you can conclude following**1st Quadrant** ==> All Functions are positive**2nd Quadrant** ==> Sin**θ** and Cosec**θ** are positive**3rd Quadrant**==> Tan**θ** and Cot**θ** are positive**4th Quadrant** ==> Cos**θ** and Sec**θ** are positive

We can also represent the coordinates with π (pi) notation

It’s been known that π = 180 degree, we express the coordinates as follows

The above illustration has expressed different coordinates in the form of π (pi)

1st Coordinates lies between –> 0 to π/2 (pi)

2nd coordinates lies between –> π/2 (pi) to π (pi)

3rd coordinates lies between –> π (pi) to 3π/2 (pi)

4th coordinates lies between –> 3π/2 (pi) to π (pi)

Let us understand the concept with the help of some examples

**Example 01 **

Find the value of sin (31π/3)

**Solution**

sin (31π/3) ==> sin (10π +π/3)

Below is the graphical representation of sin (10π +π/3)

From the above illustration we can note that;

we started from 0 and then did one complete rotation 2π to reach at same place.

We have to perform the same rotation 5 times to reach at 10π, which is again at the first quadrant.

we have got the function **sin (10π +π/3)**

From below figure you can see that the function lies in First Quadrant where all the functions are positive

Hence **sin (10π +π/3)** can be written as **sin (π/3)**

The value of **sin (π/3)** = \frac { \sqrt { 3 } }{ 2 }

**Example 02****Find the value of** sin(-\frac { 11\pi }{ 3 } ) \\\ \\

**Solution**

sin(-\frac { 11\pi }{ 3 } )\quad can\quad be\quad written\quad as\quad -sin(\frac { 11\pi }{ 3 } )\\\ \\ ==>\quad -\quad sin(4\pi -\pi /3)\\\ \\ This\quad function\quad lies\quad in\quad 4th\quad quadrant\\\ \\ ==>\quad -\quad sin(-\frac { \pi }{ 3 } )\\\ \\ ==>\quad sin(\frac { \pi }{ 3 } )\\\ \\ ==>\quad \frac { \sqrt { 3 } }{ 2 }