Evaluate: tanh(-780^{\circ})

Expression: $\tanh\left({-{780}\degree}\right)$

Use $\tanh\left({t}\right)=\frac{ {e}^{t}-{e}^{-t} }{ {e}^{t}+{e}^{-t} }$ to transform the expression

$\frac{ {e}^{-{780}\degree}-{e}^{-\left( -{780}\degree \right)} }{ {e}^{-{780}\degree}+{e}^{-\left( -{780}\degree \right)} }$

Express with a positive exponent using ${a}^{-n}=\frac{ 1 }{ {a}^{n} }$

$\frac{ \frac{ 1 }{ {e}^{{780}\degree} }-{e}^{-\left( -{780}\degree \right)} }{ {e}^{-{780}\degree}+{e}^{-\left( -{780}\degree \right)} }$

When there is a $-$ in front of an expression in parentheses, change the sign of each term of the expression and remove the parentheses

$\frac{ \frac{ 1 }{ {e}^{{780}\degree} }-{e}^{{780}\degree} }{ {e}^{-{780}\degree}+{e}^{-\left( -{780}\degree \right)} }$

Express with a positive exponent using ${a}^{-n}=\frac{ 1 }{ {a}^{n} }$

$\frac{ \frac{ 1 }{ {e}^{{780}\degree} }-{e}^{{780}\degree} }{ \frac{ 1 }{ {e}^{{780}\degree} }+{e}^{-\left( -{780}\degree \right)} }$

When there is a $-$ in front of an expression in parentheses, change the sign of each term of the expression and remove the parentheses

$\frac{ \frac{ 1 }{ {e}^{{780}\degree} }-{e}^{{780}\degree} }{ \frac{ 1 }{ {e}^{{780}\degree} }+{e}^{{780}\degree} }$

Write all numerators above the common denominator

$\frac{ \frac{ 1-{e}^{{1560}\degree} }{ {e}^{{780}\degree} } }{ \frac{ 1 }{ {e}^{{780}\degree} }+{e}^{{780}\degree} }$

Write all numerators above the common denominator

$\frac{ \frac{ 1-{e}^{{1560}\degree} }{ {e}^{{780}\degree} } }{ \frac{ 1+{e}^{{1560}\degree} }{ {e}^{{780}\degree} } }$

Simplify the expression

$\begin{align*}&\frac{ 1-{e}^{{1560}\degree} }{ 1+{e}^{{1560}\degree} } \\&\approx-1\end{align*}$