Tangent Alec Contreras
- Author:
- Alec Contreras
Why this works:
The reason we were able to use a unit circle is because we found a ratio between two triangles which contain cosine and sine.
This was done through:
The triangle that was created from O to E, from O to P and from P to E (the larger triangle)
Then we notice what was already there:
A smaller triangle: with line m:x=t going through point Q and because the ratio was one, the segment from O to Q could be seen as a hypotenuse with 1 defined as its length.
Using definitions of Sine=Opposite/Hypotenuse and Cosine=Adjacent /Hypotenuse we're able to discover the first part of the ratio as the opposite and adjacent sides defined as sin(t)/cos(t)
Because we wanted tangent the larger triangle had the length from O to P as 1. Then by definition of tangent=opposite/adjacent, we could define the segment P to E as the side that represents tan(t); and we're also able to use point E with (1,tan(t)).
Next you set up the ratio of the opposite and the adjacent sides of the smaller triangle with the larger triangle and you notice that you're left with tan(t), realizing that by moving by one unit, the height changes along with the tan(t); hence your point F is able to trace it.
Coordinates:
Q=traces the unit circle from (0,2pi)
B=(cos(pi/2), sin(pi/2)
A= represents the range from [-1,1]
Why the construction of tangent using coordinates works:
tangent is defined as: tan(q)=sin(q)/cos(q), so when q=1, tangent has an asymptote.
Construction of tangent:
Start off with the first four steps of sine in reference to the construction of sine and cosine.
This includes making a unit Circle C, with origin O (0,0) and a point Q on the circle. Make sure you create line m: x=t; line h along point Q parallel to the x-axis
line b on Point P(1,0), parallel to the y axis. Create line d from O to Q. Find the intersect between line m and line d name it F and trace it.