Trigonometric Functions

Using Euler's formula, it is possible to write the basic trig functions in terms of complex exponentials.  Cosine and sine are just the real and imaginary parts of the complex exponential

The tangent is then the ratio

The secant, cosecant, and cotangent are just the reciprocals

The angle addition formulas can be derived easily using the complex exponential forms.  First let's look at angle addition formulas for sine and cosine

noting that

the real and imaginary parts give the results

The angle difference (subtraction) formulas can be found by replacing $\beta$ with $-\beta$ in the above expressions and noting that cosine has even parity while sine has odd parity,

The addition and subtraction formulas for tangent are simply the ratios of these

The double angle formulas can be found by setting $\beta=\alpha$ in the addition formulas

The half-angle formulas can be found by using the Pythagorean identity to re-write the double angle formula for cosine in two different ways

Then substituting $\beta=2\alpha$

The the half-angle formulas are then

In general, multiple angle formulas can be computed from the real and imaginary parts of the complex exponential

thus

and

For example, the half angle formula for cosine can be computed as

squaring both sides of the equation yields

which, upon expanding the square on the right hand side and simplifying, becomes

which is the same as the result above. Using the complex form of the trigonometric functions also allows us to write explicit representations for the inverse trigonometric functions.