Recall that a triangle has three angles that add up to $180^\circ$. A right triangle is a triangle that has one $90^\circ$ angle; the side opposite this angle is called the hypotenuse. In this activity, we will look at two special cases of right triangles: when the remaining two angles are each $45^\circ$, and when the remaining two angles are $30^\circ$ and $60^\circ$. Our goal is to determine the lengths of the legs of these triangles when the hypotenuse is 1.

For this activity, you may wish to split yourself into groups to work on the various aspects of each graphing problem.

In this activity, you will discover the relationship between functions and inverse functions.

There are many angles that are not one of the standard angles on the unit circle that can, however, be written as a combination of standard angles from the unit circle. For example, $15^\circ = 45^\circ - 30^\circ$, and $75^\circ = 45^\circ + 30^\circ$. The suggests that it would be useful to derive a formula for the sine and cosine of angles that are not one of the standard angles on the unit circle. In this activity, you will apply the distance formula to some wisely-chosen pairs of points to derive formulas for $\cos (A-B)$ and $\cos (A+B)$, where $A$ and $B$ are two arbitrary angles.

In the previous activity, we derived formulas for $\cos (A+B)$ and $\cos (A-B)$ in terms of $\cos (A),\sin (A),\cos (B)$, and $\sin (B)$. In this activity, we will use those formulas to derive angle-sum and angle-difference formulas for the sine and tangent functions.

In this activity, you will discover formulas for $\sin (2\theta )$, $\cos (2\theta )$, $\sin \left (\frac {\theta }{2}\right )$ and $\cos \left (\frac {\theta }{2}\right )$, using your knowledge of the Angle Sum Formulas.

In this activity, you will discover the issue involved in using the Law of Sines and Law of Cosines to solve triangles.

In this activity, you will discover additional formulas for calculating the area of a triangle in the SAS case and in the cases when two angles and a side length are given. The SSS case can be done in a similar manner, though the most common form, called Heron’s Formula is not easily derived.

In this activity, you will discover a simple formula for the area when two of the sides of the triangle are given as vectors that begin at the origin.

In this activity, you will learn how to ‘complete the square’ of a polynomial, that is, you will learn how to modify a polynomial such that it can be written as a perfect square. This tool will allow us to rewrite polynomial equations in the standard form of circles, ellipses, and parabolas.

In this activity, you will discover formulas for the solutions of $2\times 2$ linear systems.

In this activity, you will make connections between arranging colored balls and entries in Pascal’s Triangle. By the end of this activity, you should understand the role that recursion plays in Pascal’s Triangle (e.g., how each entry depends on entries before it).