If a point on the Cartesian plane lies at (4, 2), what is the angle made between the line containing the point and the origin, and the negative y-axis?
a)
0.523 radians
b)
1.249 radians
c)
0.463 radians
d)
1.047 radians
Been stuck on this for a few days now and can't figure it out, any help would be greatly appreciated!!
AAAaannndd the bot gets it wrong yet again!
If the line makes an angle θ with the +x axis, then it makes an angle θ/2 - θ with the -y axis.
It asked for the angle between the line in quad I and the negative y-axis, so
it would be
θ + π/2 = appr 2.034 radians or appr 116.57°
To find the angle between the line containing the point (4, 2) and the origin and the negative y-axis, we can use trigonometry.
First, we need to find the slope of the line connecting the point and the origin. The slope can be calculated using the formula:
slope = (y2 - y1) / (x2 - x1)
Let's choose the origin (0, 0) as our second point, so x2 = 0 and y2 = 0. Plugging in the values, we have:
slope = (0 - 2) / (0 - 4) = -2 / -4 = 1 / 2
Next, we need to find the angle between this line and the negative y-axis. This angle can be determined using the inverse tangent function (arctan) of the slope:
angle = arctan(slope)
Using a calculator, we find that arctan(1/2) is approximately 0.464 radians.
Therefore, the answer is c) 0.463 radians.
To find the angle, we need to first find the slope of the line passing through the point (4, 2) and the origin.
The slope of a line passing through two points (x1, y1) and (x2, y2) is given by:
m = (y2 - y1)/(x2 - x1)
Using the coordinates of the given point (4, 2) and the origin (0, 0), we get:
m = (2 - 0)/(4 - 0) = 1/2
Now, we know the slope of the line passing through the point and the origin.
The angle made between this line and the negative y-axis can be found using the inverse tangent function:
tan θ = m
θ = tan^-1(m)
θ = tan^-1(1/2)
θ = 0.464 radians (approx)
Therefore, the answer is option c) 0.463 radians.