find the angle made by the line y=x with positive direction of x axis
Answers
Answer:
Step-by-step explanation:
What is the angle of a line with the Y axis and positive direction of the Y axis?
I presume that what is required is the angle that a line makes with the positive Y axis.
We shall take the positive Y axis as the initial arm and the line as the terminal arm of the angle.
Consider the line ax+by+c=0.
The slope of this line is −ab
and the angle that this line makes with the positive X axis is the inverse tangent of the slope.
Let now consider a new coordinate system where the old positive Y axis is the new positive X axis and the old negative X axis is the new positive Y axis.
In this new coordinate system the angle that the line makes with the positive X axis is the same as the angle that the line makes with the positive Y axis in the old coordinate system.
In this new coordinate system, the point (a,b)
of the old coordinate system becomes (b,−a).
⇒
In the new coordinate system, the slope of the line would be the negative inverse of the slope of the same line in the old coordinate system.
The angle that the line makes with the positive X axis in the new coordinate system is the inverse tangent of the slope in the new coordinate system.
The angle that the line ax+by+c=0
makes with the positive X axis in the old coordinate system is arctan(−ab).
⇒
The angle that the line ax+by+c=0 makes with the positive X axis in the new coordinate system is arctan(ba).
⇒
The angle that this line makes with the positive Y axis in the old coordinate system is arctan(ba).
If we write in this way, the angles would show negative values also wheres the inclination of a line is, by convention, written as positive values.
To overcome this problem, we can express the angle that this line makes with the positive Y axis as arccot(ab).
So, we conclude that the angle that a line ax+by+c=0
makes with the positive Y axis is arccot(ab).
Answer:
find the angle made by the line y=x to the positive x axis