<p class="paragraph-header">IB AA SL: core and optional content</p>
<p>Core: the sine and cosine double-angle formulae. Optional: tangent double angle and compound-angle derivations.</p>
<p>Set B = A in the addition formulae.</p>
<div class="content-box-placement content-box-placement--full content-box-placement--align-center" style="display:flex;justify-content:stretch;max-width:100%"><div class="content-box content-box--emphasis" data-content-box-preset="emphasis" style="box-sizing:border-box;border-color:var(--content-box-border, #1E6FBA);border-style:solid;border-width:0px;border-radius:0px;background-color:var(--content-box-background, #F5F8FC);color:var(--content-box-foreground, #111827);padding:12px 16px;font-family:'EB Garamond', serif;font-size:16px;line-height:1.35;width:100%;max-width:100%;border-left-width:3px"><p style="text-align: center">\(\sin2A=2\sin A\cos A\)</p></div></div>
<div class="content-box-placement content-box-placement--full content-box-placement--align-center" style="display:flex;justify-content:stretch;max-width:100%"><div class="content-box content-box--emphasis" data-content-box-preset="emphasis" style="box-sizing:border-box;border-color:var(--content-box-border, #1E6FBA);border-style:solid;border-width:0px;border-radius:0px;background-color:var(--content-box-background, #F5F8FC);color:var(--content-box-foreground, #111827);padding:12px 16px;font-family:'EB Garamond', serif;font-size:16px;line-height:1.35;width:100%;max-width:100%;border-left-width:3px"><p style="text-align: center">\(\cos2A=\cos^2A-\sin^2A=2\cos^2A-1=1-2\sin^2A\)</p></div></div>
<div class="content-box-placement content-box-placement--full content-box-placement--align-center" style="display:flex;justify-content:stretch;max-width:100%"><div class="content-box content-box--emphasis" data-content-box-preset="emphasis" style="box-sizing:border-box;border-color:var(--content-box-border, #1E6FBA);border-style:solid;border-width:0px;border-radius:0px;background-color:var(--content-box-background, #F5F8FC);color:var(--content-box-foreground, #111827);padding:12px 16px;font-family:'EB Garamond', serif;font-size:16px;line-height:1.35;width:100%;max-width:100%;border-left-width:3px"><p style="text-align: center">\(\tan2A=\dfrac{2\tan A}{1-\tan^2A}\)</p></div></div>
<p>The tangent formula requires \(\cos A\ne0\) and \(\tan^2A\ne1\).</p>
<p class="paragraph-header">Choose a useful form</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 540px;"><colgroup><col style="width: 44.4444%;"><col style="width: 55.5556%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>You want</strong></th><th style="text-align: center; --cell-justify: center"><strong>Use</strong></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">Sines only</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">Cosines only</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">Rearranged forms for proofs</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr></tbody></table></div></div>
<p class="paragraph-header">Other arguments</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 540px;"><colgroup><col style="width: 55.5556%;"><col style="width: 44.4444%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Statement</strong></th><th style="text-align: center; --cell-justify: center"><strong>Read as</strong></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">\(\sin 6\theta=2\sin 3\theta\cos 3\theta\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">\(\sin\theta=2\sin\dfrac{\theta}{2}\cos\dfrac{\theta}{2}\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">\(\cos 8\theta=1-2\sin^2 4\theta\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr></tbody></table></div></div>
<p>Doubling the angle does not generally double the ratio.</p>

Core: the sine and cosine double-angle formulae. Optional: tangent double angle and compound-angle derivations.

\(\cos2A=\cos^2A-\sin^2A=2\cos^2A-1=1-2\sin^2A\)

The tangent formula requires \(\cos A\ne0\) and \(\tan^2A\ne1\).

\(\begin{aligned} 1-\cos 2A&=2\sin^2 A \\ 1+\cos 2A&=2\cos^2 A \end{aligned}\)

\(\sin 6\theta=2\sin 3\theta\cos 3\theta\)

\(\sin\theta=2\sin\dfrac{\theta}{2}\cos\dfrac{\theta}{2}\)

\(\theta\) is double \(\dfrac{\theta}{2}\).

Doubling the angle does not generally double the ratio.

Double Angle Formulae

Double Angle Formulae

Video Solution:

Solution:

Solution locked

Statement	Read as
\(\sin 6\theta=2\sin 3\theta\cos 3\theta\)
\(\sin\theta=2\sin\dfrac{\theta}{2}\cos\dfrac{\theta}{2}\)
\(\cos 8\theta=1-2\sin^2 4\theta\)

You want	Use
Sines only
Cosines only
Rearranged forms for proofs