# Angle between function and axis

• fermio
In summary, the angle between the function y=\sqrt{3}x and the Ox axis is \frac{\pi}{3}. This can be found by calculating the tangent of the angle between the function and the x-axis, which is also the same as the tangent of the angle between the tangent line at x=0 and the x-axis. By using trigo and drawing a triangle, it can be determined that the tangent of the angle is \sqrt{3}, which leads to the angle being \frac{\pi}{3}.
fermio

## Homework Statement

What angle is between function $$y=\sqrt{3}x$$ and Ox axis?

## Homework Equations

For example is logicaly clear that angle between function y=x is 45 degrees or $$\frac{\pi}{4}$$

## The Attempt at a Solution

I just know that answer is $$\frac{\pi}{3}$$, but can't understand how to get it.

Hint: the angle between the function and the x-axis is the same angle as between the tangent line at x = 0 and the x-axis (draw a picture to see why).
Can you solve it now?

If x=0 then y=0. I can't understand. More concretly, how to calculate it?

Why do you need to set x,y=0? The question is why the angle between the line graph and the axis is pi/3, not the angle between the point (0,0) and the x-axis, which doesn't make sense. You can see that the graph is a line right? Now, let theta be the angle between the line and the x-axis. Do you know of way to find theta using trigo? You'll have to draw a triangle to see it.

The slope of a line, such as y= x, is than tangent of the angle between the line and the x-axis. As you said before, the angle between the line y= x and the x-axis is $\pi/4$. tan($\pi/4$)= 1. What is the slope of y= $\sqrt{3}$ x? What angle has that tangent?

$$\arctan\sqrt{3}=\frac{\pi}{3}$$

Defennnder said:
Why do you need to set x,y=0? The question is why the angle between the line graph and the axis is pi/3, not the angle between the point (0,0) and the x-axis, which doesn't make sense.
Sorry, I misread the question, I thought it said $$y = \sqrt{3x} = (3x)^{1/2}$$ instead of $$y = \sqrt{3}x = (3)^{1/2} \cdot x$$. I had a picture in my mind of drawing the tangent line at the origin and then calculating the angle of that with the x-axis, which could of course be done at any point. But since the function is just a straight line, it doesn't matter in this case (y' does not depend on x)

$$\tan\alpha=\frac{y}{x}=\frac{x\sqrt{3}}{x}=\sqrt{3}$$
$$\alpha=\arctan \sqrt{3}=\frac{\pi}{3}$$

Yep, that's the way to get it.

1.

## What is the angle between a function and an axis?

The angle between a function and an axis is the angle formed between the function and the x-axis or the y-axis. It measures the slope or direction of the function at a specific point.

2.

## How is the angle between a function and an axis calculated?

The angle between a function and an axis is calculated using trigonometric functions such as tangent or arctangent. It can also be calculated by finding the slope of the function at a given point and using inverse trigonometric functions to determine the angle.

3.

## What does a positive or negative angle between a function and an axis indicate?

A positive angle between a function and an axis indicates that the function is increasing or going in an upward direction. A negative angle indicates that the function is decreasing or going in a downward direction.

4.

## Can the angle between a function and an axis be greater than 90 degrees?

Yes, the angle between a function and an axis can be greater than 90 degrees. This typically occurs when the function has a steep slope or is decreasing rapidly.

5.

## How is the angle between a function and an axis used in mathematics or science?

The angle between a function and an axis is often used in calculus to find the rate of change or slope of a function. It is also used in physics to determine the direction and magnitude of forces acting on an object.

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