Find the Acute Angle Formed by Line y-(sqrt(3))x+1=0 & x-Axis

In summary, to find the acute angle formed by the line y - (sqrt(3)) x + 1 = 0 and the x-axis, we can use the fact that the gradient of the line is equal to the tangent of the angle we want to find. Simplifying the equation to y = sqrt(3)x, we can choose a point (sqrt(3), 3) on the line to form a triangle with the base of sqrt(3) and height of 3. Using SOH-CAH-TOA, we can find that the tangent of the angle is 3/sqrt(3), leading to an angle of 60 degrees or pi/3 radians.
  • #1
Agent_J
13
0
Find the acute angle that is formed by the line y - (sqrt(3)) x + 1 = 0 and the x-axis.

better picture here:
http://members.rogers.com/agentj/images/math.jpg

I am totally lost with this one. It was from my old trigonometry test, but I don't see the relavance of the question to trigonometry :frown:
 
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  • #2
you know the gradient and that's tan of the angle you want
 
  • #3
so how would I start tackling this question :redface: ?
 
  • #4
What this is basically is polar coordinates.

We solve for y, and get y = sqrt(3) - 1. It's easier to do if the graph intersects the origin. Because this is just a straight line, the -1 can be removed from the equation and it won't change the angle to the x-axis. So, our equation is...

y = sqrt(3)x

Next, we need to get a point. Just to keep nice even answers, I'll use x = sqrt(3).

y = sqrt(3)x
y = sqrt(3)*sqrt(3)
y = 3

So, a point is (sqrt(3), 3).

Now, what you do is basically, it makes a triangle. The base (x) is sqrt(3), and the height (y) is 3.

We use our SOH-CAH-TOA trig functions, and see that tan = opp/adj. The opposite angle is the height/y, and the adjacent angle is the base/x.

I'm just using the letter t for now to make things easier...

tan t = 3/sqrt(3)
t = arctan(3/sqrt(3))

t = 60 degrees, or pi/3 if you're working in radians

NOTE: I always seem to goof something up whenever I try to help here, so someone else should just double check what I did.
 
  • #5
apart from the fact that you made it far more complicated than it needs to be, that is correct.

y=mx+c

then the gradient is m and that is tan of the angle of the slope, that's all.
 

1. What is an acute angle?

An acute angle is an angle that measures less than 90 degrees. It is typically associated with a sharp or narrow angle.

2. How do I find the acute angle formed by a line and the x-axis?

To find the acute angle, you will need to use the slope of the line. First, solve the equation of the line for y and write it in slope-intercept form (y=mx+b). The slope (m) will be the tangent of the angle formed by the line and the x-axis. Find the inverse tangent of the slope to get the angle in radians. Finally, convert the angle to degrees by multiplying by 180/pi.

3. What is the equation of the line y-(sqrt(3))x+1=0?

The given equation is in point-slope form, where the slope is -(sqrt(3)) and the y-intercept is 1. To put it in slope-intercept form, you would solve for y, which would give you y=(sqrt(3))x+1.

4. How do I know if the angle formed by the line and the x-axis is acute?

You can determine the acute angle by finding the slope of the line. If the slope is positive, the angle is acute. If the slope is negative, the angle is obtuse (greater than 90 degrees). If the slope is zero, the angle is a right angle (90 degrees).

5. Can I use a calculator to find the acute angle?

Yes, you can use a calculator to find the inverse tangent of the slope of the line to get the angle in radians. From there, you can convert to degrees by multiplying by 180/pi. However, it is always a good idea to understand the steps and concepts behind the calculation rather than relying solely on a calculator.

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