Mr. Giuseppe Cammarata at the Liceo Scientifico Statale Benedetto Croce in Palermo, Italy is using this material in his Course IV F. Many of the solutions below were written by his students.

7. We use Euler's formula, to show the truthfulness of the Pythagorean Identity. We find the conjugate for Euler's formula and we multiply the two equations.

q.e.d.

Solution by Giuseppe Messineo and Raffaele Iacobucci
Liceo Scientifico Statale Benedetto Croce, May 2001.

13. Double angles formulas for sine and cosine - see Example 1.

Addition formulas for sine and cosine - see Example 2.

Subtraction formulas for sine and cosine.

First we note that It follows for the Euler formula that:

But we can also write Equating the real and the immaginary parts of these equations, we have:

Subtraction formula for tangent - see Example 3.

Double angle formula for tangent.

We have seen (in Example 3) that Now we can write:

It follows from the definition of the tangent:

Addition formula for tangent.

We have seen (in Example 3) that Now we perform multiplication between this and another one, and get

Now it follows from the definition of the tangent that:

Subtraction formula for cotangent.

We have seen that from the Euler formula. Another way to write it is when we consider the magnitude as

Taking the conjugates of another complex number and multipling, we have:

Now it follows from the definition of the cotangent that:

Addition formula for cotangent.

We multiply as above to obtain:

Now it follows from the definition of the cotangent that:

Double angle formula for cotangent.

We multiply as above to obtain:

It follows from the definition of the cotangent:

Solution by Marco Modica and Fabrizio Martino
Liceo Scientifico Statale Benedetto Croce, May 2001.

15 (a)

(b) We observe angle x individualized by point P and let -P be its negative. The argument of -P is Then, if P = (z, y) and -P = (z', y'), we can write:

and we obtain:

Finally, since

(c)

(d)

Solution by Anna La Rocca and Marianna Bruno
Liceo Scientifico Statale Benedetto Croce, May 2001.

19. Construct on the Argand plane a circle having r = hyp and center coinciding with the vertice of see the figure. We note that:

r = hyp is the magnitude of point B;

a = adj is the real part of point B;

b = opp is the imaginary part of point B.

Using the definitions of the trigonometric functions, we have:

Solution by Laura Marretta and Candido Zacchino
Liceo Scientifico Statale Benedetto Croce, May 2001.