Angle made by the line y = -√3x + 3 with positive direction of x-axis

To find the angle made by the line with the positive direction of the x-axis, we can use the slope of the line. The slope of a line is given by the ratio of the change in the y-coordinates to the change in the x-coordinates between two points on the line. In this case, since we have the equation of the line, we can determine the slope directly.

The equation of the line is given as y = -√3x + 3. We can rewrite it in the form y = mx + b, where m is the slope of the line. Comparing the equation with the standard form, we can see that the slope is -√3.

Calculating the angle:

The angle made by the line with the positive direction of the x-axis can be found using the tangent function. The tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

In this case, the slope of the line (-√3) represents the tangent of the angle we are looking for. The tangent of an angle can be determined using a calculator or a table of trigonometric values. Taking the inverse tangent (arctan) of the slope (-√3) will give us the angle made by the line with the positive direction of the x-axis.

Using the inverse tangent function:

arctan(-√3) ≈ -60°

Interpreting the angle:

The angle we found is approximately -60°. Since the angle is negative, it means that the line makes a clockwise rotation from the positive direction of the x-axis.

Summary:

The line y = -√3x + 3 makes an angle of approximately -60° with the positive direction of the x-axis. This means that the line rotates clockwise from the positive x-axis direction.

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