Imaginary Numbers and Trigonometry

In our earlier discussion of imaginary numbers, we learned how a picture of the complex number two plus three i can be drawn.

If we draw a right triangle in the picture, then we have:

a right triangle with vertices at the origin, at x equals 2, and at the complex number 2 plus 3 i

The parts of the complex number two plus three i are shown by the horizontal and vertical sides of the triangle.  The hypotenuse of the right triangle has length square root of 13, found by using the Pythagorean Theorem or the distance formula.

In the triangle, if we use P for the name of the angle at the origin, then we can rewrite the number two plus three i by using its trigonometric components.  The horizontal component is the square root of 13, times the cosine of P, the vertical component is the square root of 13, times the sine of P, times i, so the number two plus three i is the same as the square root of 13, times the quantity cosine P plus i times sine P.  Basically, every complex number can be written in the form r times the quantity cosine P plus i times sine P.

Multiplying and dividing

There are many identities in trigonometry, and they are the key to multiplying and dividing complex numbers.  Suppose we have the two complex numbers r times the quantity cosine P plus i times sine P and s times the quantity cosine Q plus i times sine Q.  When we multiply them using distributive property, we obtain cosine P cosine Q, plus i cosine P sine Q, plus i sine P cosine Q, minus sine P sine Q, all times r s.  But the formulas for the sine and cosine of a sum allow us to rewrite this quantity as cosine of P plus Q, plus i times the sine of P plus Q, all times r s.  In words, multiplying complex numbers involves multiplying their distances from the origin, and adding their angles.

We can derive a similar formula for dividing complex numbers.  That formula is cosine P minus Q, plus i times sine of P minus Q, all times r over s.

Powers and roots

Since exponents represent repeated multiplication, we can derive a formula for powers of complex numbers.  That formula is known as DeMoivre's Theorem:

the nth power of the number r times the quantity cosine P plus i sine P equals the nth power of r times cosine of n P plus i times sine of n P

We can also use this formula to do roots, by using fractions for the variable n.  However, roots are not unique, since the trigonometric functions are periodic.  The formula which gives the nth roots of r times the quantity cosine P plus i times sine P isformula for roots of a complex number.  Notice that this formula not only does square roots, but all kinds of roots.

So what is the square root of 2 plus 3 i?  In trigonometric form, the original number is cosine P plus i sine P, all times the square root of 13, where the angle P equals the arctangent of three halves.  Therefore, the square roots arecosine of one half the arctangent of three halves, plus i times the sine of one half the arctangent of three halves, all times the fourth root of 13, and as before, but with an angle whose value is pi larger.

Epilogue

Of course, now we can simply ask some hand held calculators to do our complex number arithmetic.  Unfortunately, the calculator doesn't tell us why the answers are the way they are.  This discussion was intended to help our understanding.

Now, if you have mastered square roots of imaginary numbers, are you ready for imaginary numbers as exponents?  If so, read Using i as an Exponent.

Back to introduction.