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\noindent
{\large\bf Trigonometric Identities:}

When solving equations that involve trigonometric
functions, one needs to know the relationships
between the different functions. 
We have already seen a number of trig identities:\footnote{An 
{\em identity} is an equation valid under all 
allowable substitutions.}

\bigskip

\noindent
{\bf Defining identities:}

\[\tan(x) = \frac{\sin(x)}{\cos(x)},\;\;
\sec(x) = \frac{1}{\cos(x)},\;\;
\csc(x) = \frac{1}{\sin(x)},\;\;
\cot(x) = \frac{1}{\tan(x)} = 
\frac{\cos(x)}{\sin(x)}.\]

\bigskip

\noindent
{\bf Pythagorean Theorem:} 
\[\cos^2(x) + \sin^2(x) = 1, \;\;
1 + \tan^2(x) = \sec^2(x),\;\;\mbox{\rm or}\;\;
\cot^2(x) + 1 = \csc^2(x).\]


\bigskip

\noindent
{\bf Cofunction identities:}
\[FUNCTION\left(\frac{\pi}{2}-x\right) = COFUNCTION(x).\]
The pairs of cofunctions are 
$\{\mbox{\rm sine, cosine}\}$, 
$\{\mbox{\rm tangent, cotangent}\}$, and 
$\{\mbox{\rm secant, cosecant}\}$.
So $\displaystyle 
\sin\left(\frac{\pi}{2}-x\right) = \cos(x)$ and 
$\displaystyle 
\cos\left(\frac{\pi}{2}-x\right) = \sin(x)$, etc.

\bigskip


\bigskip

Now we look at some new identities.
The first, which is the `grandaddy' of all trigonometric
identities, relates the trig functions to 
the exponential function. It is this theorem
which explains why the base $e$ is the most 
useful exponential base in mathematics.

\bigskip

\noindent
{\bf Euler's Theorem:}
\[e^{a+bi} = e^a\left(\cos(b)+i\sin(b)\right).\]

\bigskip

\noindent
{\bf Consequences of Euler's Theorem:}

\medskip

$\bullet$ (Euler's Formula) Set $a+bi = 0+\pi i$.

\smallskip

$\displaystyle
e^{\pi i} + 1 = 0$ 

\vspace{.4in}

$\bullet$ (Pythagorean Theorem) Use $e^{ix}\cdot e^{-ix} = e^0 = 1$.
\smallskip

$\displaystyle
\cos^2(x) + \sin^2(x) = 1$ 

\vspace{.4in}

$\bullet$ (Cofunction Identity) Use $e^{\left(\frac{\pi}{2}-x\right)i} = 
e^{\frac{\pi}{2}i}e^{-xi}$.

\smallskip

$\displaystyle
\sin\left(\frac{\pi}{2}-x\right) = \cos(x)$

\vfill

\break

\noindent
Now we look at some new identities that are consequences
of Euler's Theorem.

\bigskip

$\bullet$ (Angle Addition Formulas) Use $e^{(x\pm y)i} =
e^{ix}e^{\pm iy}$.

\smallskip

$\displaystyle
\sin(x\pm y) = \sin(x)\cos(y) \pm \sin(y)\cos(x)$

\smallskip

$\displaystyle
\cos(x\pm y) = \cos(x)\cos(y) \mp \sin(y)\sin(x)$

\smallskip

$\displaystyle
\tan(x\pm y) = \frac{\tan(x)\pm\tan(y)}{1\mp \tan(x)\tan(y)}$

\vspace{.5in}

$\bullet$ (Product-Sum Formulas) 

\smallskip

$\displaystyle
\sin(x)\cos(y) = \frac{1}{2}\left(\sin(x+y) + \sin(x-y)\right)$


\smallskip

$\displaystyle
\sin(x)\sin(y) = \frac{1}{2}\left(\cos(x+y) - \cos(x-y)\right)$


\smallskip

$\displaystyle
\cos(x)\cos(y) = \frac{1}{2}\left(\cos(x+y) + \cos(x-y)\right)$


\vspace{.5in}

$\bullet$ (Double Angle Formulas) Set $x=y$ in the Angle 
Addition Formulas.

\smallskip

$\displaystyle
\sin(2x) = 2\sin(x)\cos(x)$

\smallskip

$\displaystyle
\cos(2x) = \cos^2(x) - \sin^2(x) = 1-2\sin^2(x) = 2\cos^2(x) - 1$

\vspace{.5in}

$\bullet$ (Half Angle Formulas) 

\smallskip

$\displaystyle
\sin\left(\frac{x}{2}\right)  = \sqrt{\frac{1-\cos(x)}{2}}$

\smallskip

$\displaystyle
\cos\left(\frac{x}{2}\right) = \sqrt{\frac{1+\cos(x)}{2}}$

\vspace{.5in}

$\bullet$ (Polar Form for Complex Numbers) 
If $z$ is a complex number, then 
\[z = |z|(\cos(x) + i\sin(x))\]
for some $x$.
Moreover, if $w = |w|(\cos(y) + i\sin(y))$,
then $zw = |zw|(\cos(x+y) + i\sin(x+y))$.

\bigskip

\vspace{.5in}

$\bullet$ (De Moivre's Theorem) Use $\left(e^{ix}\right)^n = e^{inx}$.
\[(cos(x)+i\sin(x))^n = \cos(nx) + i\sin(nx).\]

\bigskip

\break

\noindent
{\bf Exercises.}

\begin{enumerate}
\item[(1)] Find $\displaystyle \cos\left(\frac{\pi}{12}\right)$
using the fact that $\displaystyle 
\frac{\pi}{12} = \frac{\pi}{3} - \frac{\pi}{4}$.
\item[(2)] Find $\displaystyle \cos\left(\frac{\pi}{12}\right)$
using the fact that $\displaystyle 
\frac{\pi}{12} = \frac{1}{2}\cdot \frac{\pi}{6}$.
\item[(3)] Did you get the same answer in (1) and (2)?
\item[(4)] Rewrite $\sin^4(x)$ in a way that does not 
involve products of trig functions.
\item[(5)] Rewrite $\cos(x)\sin^3(x)$ in a way that does not 
involve products of trig functions.
\item[(6)] Explain why $\displaystyle \sqrt{i} =
\pm\left(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i\right)$.
\item[(7)] Find $\sqrt[3]{i}$.
\end{enumerate}


\end{document}
