What are you allowed to use here? For instance, are you allowed to use that tan is defined as sin/cos, and that sin is opposite/hypotenuse and cos is adjacent/hypotenuse?
Given the following naming convention:
It doesn't seam trivial to me to prove that the length of the segment named "tan \theta " is in fact equal to opposite / adjacent, where opposite and adjacent refer to the sides of the triangle inside the circle.
Can any help with this?
The title of the thread might be a little confusing. What I am trying to prove here is that the length of the segment named "tan \theta" is actually the same as the value of the trigonometric function tangent(angle).
By now I actually think I solved the problem though:
Given the tangent definition "tan = sin / cos" and known that sin = oppos / hyp and cos = adj / hyp -> tan = opposite / adjacent. Now my real question was "How do we know that the length of the segment is equal to oppos / adj" The answer is straight forward if we give a meaning to each side of the triangle. the adjacent side can be thought as the distance covered along the x axis, the opposite side is a difference of height between the ends of the hyp segment, and hypotenuse is therefore the slope which is deffined as height / distance.
Finally since the radius of the circle is 1 we can assert that the the length of the segment named "tan \theta" is actually equal to the trigonometric function tangent(\theta )
Does it make sense?
Thanks for the reply
Using the unit circle, you explain that the right-angle triangle that is formed by the lengths is similar to the right-angle triangle formed by the lengths .
That means and , where is some constant.
From the second equation, we can see and so , or .
Now remembering that and , that means
I can't see the pictures but I suspect you have a unit circle (center at (0, 0), radius 1) with a line through the origin at angle and a vertical line tangent to that circle at (1, 0). Yes, the length of the segment from (1, 0) to the original line is .
The point at which the line crosses the circle has coordinates (that's one common way of defining sine and cosine). That is, the slope of the line is . But then, of course, the slope as calculated by using (1, y) (the point where the vertical line at (1, 0) crosses that line) and (0, 0) is .