Find the equation of the line in the xy-plane that goes through the origin and makes an angle of 1.2 radians with the positive x-axis.
that would be
y = tan(1.2) x
To find the equation of the line in the xy-plane, we can start by using the fact that two points determine a line. We are given that the line passes through the origin, which means we have one point (0, 0).
Next, we need to find a second point on the line. We know that the line makes an angle of 1.2 radians with the positive x-axis. By considering the unit circle, we can determine the coordinates of a second point on the line.
The unit circle is a circle with a radius of 1. Since the angle between the positive x-axis and the line is 1.2 radians, we can use trigonometry to find the coordinates of a point that lies on the line.
The x-coordinate of this point can be found using the cosine function. Since the angle is measured from the positive x-axis, the cosine of the angle will give us the x-coordinate.
To find the y-coordinate, we can use the sine function. Again, since the angle is measured from the positive x-axis, the sine of the angle will give us the y-coordinate.
Using these trigonometric functions, we can find the x and y-coordinates of the second point on the line. The coordinates can be written as (x, y), where x is the x-coordinate and y is the y-coordinate.
Once we have the coordinates of the second point, we can use the equation of a line to find the equation in the form y = mx + b, where m is the slope and b is the y-intercept.
The slope (m) of the line can be found using the formula:
m = (y2 - y1) / (x2 - x1),
where (x1, y1) are the coordinates of the first point (the origin) and (x2, y2) are the coordinates of the second point on the line.
Since the line passes through the origin, the y-intercept (b) will be 0. Therefore, the equation of the line will be of the form y = mx.
Let's calculate the coordinates of the second point using the angle of 1.2 radians:
x = cos(1.2) ≈ 0.36237
y = sin(1.2) ≈ 0.93204
Now, we can find the slope (m) using the formula:
m = (0.93204 - 0) / (0.36237 - 0) = 2.56715
Finally, we can write the equation of the line in the xy-plane that goes through the origin and makes an angle of 1.2 radians with the positive x-axis:
y = 2.56715x