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 number
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 number
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 ==> Sinθ and Cosecθ are positive
3rd Quadrant==> Tanθ and Cotθ are positive
4th Quadrant ==> Cosθ and Secθ 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 }
