Name ____________________________________
 From Euclid to Spigotry
Assignment 4 (outline 4)
Last modified: Sept. 15, 2004
You can do everything on this piece of paper and hand it in. 
Shown in the diagram below is a triangle created by plotting the complex number z = 3 + 4i.

1. Use the half-angle formulas to compute the following (take positive square roots):

sin q/2 =                          cos q/2 =                               tan q/2 =

cot q/2 =                         sec q/2 =                                     csc q/2 =

Now find a complex number w =   which, when plotted in the first quadrant, makes an angle of q/2 with the x axis, and add it to the diagram. Also plot the other complex number that satisfies
w2 = z. 

2. Express tan (x/2) and cot (x/2) in terms of cot x and csc x. Check your answers by confirming that tan (x/2) cot (x/2) = 1.

3. Find formulas for tan (x + p/4) and for tan (x - p/4) in terms of tan x. Then show that
tan (2x) = tan (x + p/4) - tan (x - p/4). This identity is the key to constructing a regular heptadecagon (17-sided polygon).
4. By using DeMoivre's theorem, find an expression for sin 4q in terms of sin q and cos q and one for cos 4q in terms of powers cos q alone.  (The hard part may be using sin2 q + cos2 q = 1 to express your second answer in terms of cos q alone.)   Then convert the equation cos 4q = -1 into a fourth-degree equation for x = cos q and show that the solutions are what you would expect on the basis of geometry.  The obvious solution corresponds to 4q = p, but there are three other possibilities like 4q = 3p.