Introduction:
In this article we shall see about terminal point on unit circle.Unit circle is a circle with the radius and centered at the origin in the xy plane that means center point is (0,0) .For any real number t, let P(x, y) be the point on the unit circle that is a distance t from (1, 0) in counterclockwise direction if t > 0 and in the clockwise direction if t < 0. We call the point P(x,y) the terminal point determined from t.Simply we say that the terminal point is the point reached after traveling t units around the circle.
Unit Circle
unit circle
A point is on the unit circle if it satisfies that the equation of the circle, x2 + y2 = 1.
For example, (`3/5` ,`4/5` ) is on the unit circle since,
`(3/5)^(2)` +`(4/5)^(2)`
=1
The x-axes and y-axes divide the plane into four quadrants that is I, II, III, and IV, where each point (x, y) in the coordinate plane either lies on any one of the coordinate axes or any one of the four quadrants:
Quadrant I:
In this, both of the x and y are positive.
Quadrant II:
In this, x is negative and then y is positive.
Quadrant III:
In this,both of the x and y are negative.
Quadrant IV:
In this, x is positive and then y is negative.
Locating Terminal Points on the Unit Circle:
We all know the circumference of the circle is 2`pi`
Using that the following four rules are derived. we will use these rules for calculate the terminal points<br>
Separate the circle into 4 parts to calculate the terminal points determined by the angles 0, `pi/2`,`pi` ,`3pi/2` and 2`pi `
Separate the circle into 8 parts to calculate the terminal points determined by the angles `pi/4, “3pi/4` ,`5pi/4`,`7pi/4`
Separate the circle into 6 parts to calculate the terminal points determined by the angles `pi/3`,`2pi/3`,`4pi/3`,`5pi/3`
Separate the circle into 12 parts to calculate the terminal points determined by the angles `pi/3`,`5pi/6`,`7pi/6`,`11pi/6`
Finding the Coordinates of terminal points in Quadrant I:
Terminal point p(x,y) determined by the angle `pi/4 `
In this angle,the point is at the intersection of the unit circle and the line y = x. Since
x2 + y2 = 1 and x = y,
it follows that 2×2 = 1
x=`sqrt(2)/2` y=`sqrt(2)/2`
( `sqrt(2)/2`, `sqrt(2)/2`
Terminal point determined by the angle `pi/6`
In this angle,the point (x; y) is the same distance from (x,-y) as it is from the point (0; 1). Therefore,
2y =`sqrt(x^(2)+(y-1)^(2))` and so y=`1/2`
4y2=x2+(y-1)2
4.`(1/4)` =x2 +`1/4`
x2 =`3/4`
x = `(sqrt(3))/(2)`
( `sqrt(3)/2`,`1/2` )
Terminal point determined by the angle `pi/3`
In this angle the terminal point determined by `pi/3` is the reflection of the terminal point determined by `pi/6` a cross the line y = x.so here the x and y values will be interchanged.
Example:
The point P is on the unit circle. Find the terminal point P (x, y) from the following given information.
The x-coordinate of P is `1/2` and P is in quadrant I.
solution:
`(1/2)^(2)` +y2 =1
y2 = 1-`(1/4)`
y2 = `3/4`
y =± `sqrt(3)/2`
y =`sqrt(3)/2` because P is in quadrant I;
so, terminal point is (`1/2“sqrt(3)/2`)
I am planning to write more post on arc length of a circle and its example and, Area Of Sector Of A Circle and its problem with solution. Keep checking my blog.
Finding the Coordinates of Terminal Points in any Quadrant:
We will consider that P(x , y) as a terminal point determined by any value of t to find out the correct values of the terminal point.The following four rules will shows how to calculate the terminal point on unit circle in all of the quadrants of the circle:
(i) Place the point P(x, y) on the unit circle.
(ii) Then calculate the reference number t. This is the smallest distance along the unit circle between the
terminal point determined by t and the x-axis . The reference number value is always positive.
(iii) calculate exact terminal point that is determined by t.
(iv) Modify the signs of the coordinates according to the quadrant that contains the point P(x,y).
Terminal points on unit circle in all quadrants
This diagram has been drawn using the above four rules.
unit circle with terminal point in all quadrants
From this diagram we can able to know the terminal points in all quadrant of the unit circle.