# Tangent and Secant

- Author:
- John Williams

## Instructions

In the applet below, you'll find a slider for an angle . Point A is the terminal point of the angle on the unit circle.
You'll want to pause the rotation of angle to work on the applet. Click the pause button next to the slider on the left.
Use the Ray Tool to create a ray from the center B to point A. In the Input box at the bottom of the list, enter the line . Use the Point tool to create a point where your ray intersects . This point will be labeled point D.
As you saw in the Unit Triangle task on Tuesday, both the tangent and secant can be formed from this triangle. The tangent (opp/adj) is the length of the opposite side (since the adjacent side is on the unit circle), and the secant (hyp/adj) is the length of the hypotenuse.
Create a point by entering . Turn on trace for the point. Press play on .

## Tangent Curve

You'll notice that whenever point A is in quadrants II or III, point D is undefined. This is because we need to create another line for the unit triangles when point A has a negative x-value.
Press pause on and move the slider so that point A is in either quadrant II or III.
In the input box at the bottom of the list of text, input .
Use the Point tool to create a point where ray AB meets . This point will be labeled point F.
Create a point by entering . Turn on trace for the point. Press play on . Now you should have a full trace of the tangent curve.

## Secant Curve

Work on the secant curve in the workspace below, so you don't confuse the secant and the tangent curves.
To create a trace of the secant curve, you'll need to use the same set up as the tangent curve.
Use the Ray tool to create a ray from the center B to the rotating point A, the hypotenuse of your triangle.
Enter and , the adjacent sides of your unit triangles. Create a point D at the intersection of ray BA and x=1, and then slide to rotate point A so you can create a point E at the intersection of ray BA and x=-1.
Since , the length of the hypotenuse in the unit triangle will be the output of the secant curve. The length of the hypotenuse is not a coordinate of the points on the unit circle, so we can't use the method we used in the tangent curve.
Click the gray circle next to Ray(B,A) on the list to the left to remove the ray from the diagram. Use the Segment tool to create a segment from point B to E. The length of the segment should show up in the list to the left as 'i : Segment (B,E)'. If your segment shows up as a letter other than 'i', that's fine.
Move the slider for back so you that point D shows up again. Use the Segment tool to make a segment from B to D.
Create a point by entering and turn on the trace for the point. Move the slider for back so that point E shows up again. Create a point by entering and turn on the trace for the point.
Press play on the slider for .

## Analyze

Why did we have to use a negative for the second tracer point on the secant and tangent curve?

## Compare and Contrast

Compare and contrast the tangent curve and the secant curve. How are they similar? How are they different? Include an analysis of domain, range, maxima/minima, and asymptotes.

## Tough Question...

Try entering . Why can't you see this point moving as rotates?