Multiples And Sub-Multiples
Theorem : When cos2A is known, then the value of sinA, cosA tan A and cotA in terms of cos2A
i) \(
\sin A = \pm \sqrt {\frac{{1 - \cos 2A}}
{2}}
\)
We know,
cos2A = 1- 2 sin2A
\(
\Rightarrow 2\sin ^2 A = 1 - \cos 2A
\)
\(
\Rightarrow \sin ^2 A = \frac{{1 - \cos 2A}}
{2}
\)
\(
\Rightarrow \,\,\,\sin A = \pm \sqrt {\frac{{1 - \cos 2A}}
{2}}
\) ......(1)
ii) \(
\cos A = \pm \sqrt {\frac{{1 + \cos 2A}}
{2}}
\)
We know
cos2A = 2cos2A - 1
\(
\Rightarrow \,\,\,\,2\cos ^2 A = 1 + \cos 2A
\)
\(
\Rightarrow \,\,\,\,\cos ^2 A = \frac{{1 + \cos 2A}}
{2}
\)
\(
\Rightarrow \,\,\,\,\cos A = \pm \sqrt {\frac{{1 + \cos 2A}}
{2}}
\)
iii) \(
\tan A = \pm \sqrt {\frac{{1 - \cos 2A}}
{{1 + \cos 2A}}}
\)
we know \(
\tan A = \frac{{\sin A}}
{{\cos A}}
\)
\(
= \pm \frac{{\sqrt {\frac{{1 - \cos 2A}}
{2}} }}
{{\sqrt {\frac{{1 + \cos 2A}}
{2}} }}
\)
\(
\tan A = \pm \sqrt {\frac{{1 - \cos 2A}}
{{1 + \cos 2A}}}
\)...... (iiii)
iv) We know
\(
\cot A = \frac{1}
{{\tan A}}
\)
\(
\cot A = \pm \sqrt {\frac{{1 + \cos 2A}}
{{1 - \cos 2A}}}
\)......(iv)
Note : Put \(
A = \frac{A}
{2}
\) in all above formulas
i)\(
\sin \frac{A}
{2} = \pm \sqrt {\frac{{1 - \cos A}}
{2}}
\)
ii)\(
\cos \frac{A}
{2} = \pm \sqrt {\frac{{1 - \cos A}}
{2}}
\)
iii)\(
\tan \frac{A}
{2} = \pm \sqrt {\frac{{1 - \cos A}}
{{1 + \cos A}}}
\)
iv)\(
\cot \frac{A}
{2} = \pm \sqrt {\frac{{1 + \cos A}}
{{1 - \cos A}}}
\)