Half Angle Formulas

Theorem

Half Angle Formula for Sine

\(\ds \sin \frac \theta 2\)

\(=\)

\(\ds +\sqrt {\frac {1 - \cos \theta} 2}\)

for $\dfrac \theta 2$ in quadrant $\text I$ or quadrant $\text {II}$

\(\ds \sin \frac \theta 2\)

\(=\)

\(\ds -\sqrt {\dfrac {1 - \cos \theta} 2}\)

for $\dfrac \theta 2$ in quadrant $\text {III}$ or quadrant $\text {IV}$

Half Angle Formula for Cosine

\(\ds \cos \frac \theta 2\)

\(=\)

\(\ds +\sqrt {\frac {1 + \cos \theta} 2}\)

for $\dfrac \theta 2$ in quadrant $\text I$ or quadrant $\text {IV}$

\(\ds \cos \frac \theta 2\)

\(=\)

\(\ds -\sqrt {\frac {1 + \cos \theta} 2}\)

for $\dfrac \theta 2$ in quadrant $\text {II}$ or quadrant $\text {III}$

Half Angle Formula for Tangent

\(\ds \tan \frac \theta 2\)

\(=\)

\(\ds +\sqrt {\dfrac {1 - \cos \theta} {1 + \cos \theta} }\)

for $\dfrac \theta 2$ in quadrant $\text I$ or quadrant $\text {III}$

\(\ds \tan \frac \theta 2\)

\(=\)

\(\ds -\sqrt {\dfrac {1 - \cos \theta} {1 + \cos \theta} }\)

for $\dfrac \theta 2$ in quadrant $\text {II}$ or quadrant $\text {IV}$

where $\tan$ denotes tangent and $\cos$ denotes cosine.

When $\theta = \paren {2 k + 1} \pi$, $\tan \dfrac \theta 2$ is undefined.

Half Angle Formula for Tangent: Corollary 1

$\tan \dfrac \theta 2 = \dfrac {\sin \theta} {1 + \cos \theta}$

Half Angle Formula for Tangent: Corollary 2

$\tan \dfrac \theta 2 = \dfrac {1 - \cos \theta} {\sin \theta}$

Half Angle Formula for Tangent: Corollary 3

$\tan \dfrac \theta 2 = \csc \theta - \cot \theta$

One Plus Tangent Half Angle over One Minus Tangent Half Angle

$\dfrac {1 + \tan \frac x 2} {1 - \tan \frac x 2} = \sec x + \tan x$

Let $x \in \R$.

Then:

Half Angle Formula for Hyperbolic Sine

\(\ds \sinh \frac x 2\)

\(=\)

\(\ds +\sqrt {\frac {\cosh x - 1} 2}\)

for $x \ge 0$

\(\ds \sinh \frac x 2\)

\(=\)

\(\ds -\sqrt {\dfrac {\cosh x - 1} 2}\)

for $x \le 0$

Half Angle Formula for Hyperbolic Cosine

$\cosh \dfrac x 2 = +\sqrt {\dfrac {\cosh x + 1} 2}$

Half Angle Formula for Hyperbolic Tangent

\(\ds \tanh \frac x 2\)

\(=\)

\(\ds +\sqrt {\frac {\cosh x - 1} {\cosh x + 1} }\)

for $x \ge 1$

\(\ds \tanh \frac x 2\)

\(=\)

\(\ds -\sqrt {\frac {\cosh x - 1} {\cosh x + 1} }\)

for $x \le 1$

Half Angle Formula for Hyperbolic Tangent: Corollary 1

$\tanh \dfrac x 2 = \dfrac {\sinh x} {\cosh x + 1}$

Half Angle Formula for Hyperbolic Tangent: Corollary 2

For $x \ne 0$:

$\tanh \dfrac x 2 = \dfrac {\cosh x - 1} {\sinh x}$

Also known as

Some sources hyphenate: half-angle formulas.

The British English plural is formulae.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): half-angle formula
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): half-angle formula