This is a diagram of this problem
The word Problem:
Andrew wants to construct a new stairway to the front door of his house. The door is 3 feet above the ground and at an angle of $30^{\circ}$ to the sidewalk, as shown in the diagram. Let’s help him find how far from the house he needs to begin the stairway. The stairway will make an angle of $30^{\circ}$ with the ground. The angle between the ground and the front of the house is $90^{\circ}$. So, we have a $30^{\circ}-60^{\circ}-90^{\circ}$ triangle. We know the angle and the side opposite to it. We need to find the adjacent side. We know that in a right triangle, the tangent function is the ratio of the opposite side to the adjacent side. So we can use that ratio to find the adjacent side:
What I know how to do:
I know that I have to use the trigonometric function tangent to solve this problem because I am only given the opposite side and I need to solve for the adjacent side of the staircase.
tan $30^{\circ} = 3/x$
So I solve for x and get
$x = 3/x$
The notes that this problem is based on already gives the answer to the problem but I am stuck on this part of the explanation where its showing how to solve for x. This is where I am stuck
I don't understand how $\tan 30^{\circ} = 1/\sqrt{3}$.
Is it radians or am I missing something?
This is a $30^{\circ}$-$60^{\circ}$-$90^{\circ}$ triangle, so the sides are proportional to $\frac12,\frac{\sqrt{3}}{2},1$, right?
In particular, $\sin 30^{\circ} = \frac12$ and $\cos 30^{\circ}=\frac{\sqrt{3}}{2}$, so $\tan 30^{\circ} =\frac{\sin 30^{\circ}}{\cos 30^{\circ}}= \frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}}$.