<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">\(\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B\)</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">\(\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B\)</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">\(\tan(A\pm B)=\dfrac{\tan A\pm\tan B}{1\mp\tan A\tan B}\)</p></div></div>
<p>Read upper signs together and lower signs together. The tangent formula requires all displayed terms to be defined and the denominator to be nonzero.</p>
<p class="paragraph-header">Exact values</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 540px;"><colgroup><col style="width: 33.3333%;"><col style="width: 66.6667%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Split the angle</strong></th><th style="text-align: center; --cell-justify: center"><strong>Working</strong></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">\(105^\circ=60^\circ+45^\circ\)</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">\(75^\circ=45^\circ+30^\circ\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr></tbody></table></div></div>
<p>For given ratios, use the quadrants to find the signs of any missing ratios before substitution.</p>
<p>In general, \(\sin(A+B)\ne\sin A+\sin B\).</p>

\(\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B\)

\(\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B\)

\(\tan(A\pm B)=\dfrac{\tan A\pm\tan B}{1\mp\tan A\tan B}\)

Read upper signs together and lower signs together. The tangent formula requires all displayed terms to be defined and the denominator to be nonzero.

\(\begin{aligned} \sin 105^\circ&=\sin 60^\circ\cos 45^\circ+\cos 60^\circ\sin 45^\circ \\ &=\dfrac{\sqrt{3}}{2}\cdot\dfrac{1}{\sqrt{2}}+\dfrac{1}{2}\cdot\dfrac{1}{\sqrt{2}} \\ &=\dfrac{\sqrt{6}+\sqrt{2}}{4} \end{aligned}\)

\(\begin{aligned} \cos 75^\circ&=\cos 45^\circ\cos 30^\circ-\sin 45^\circ\sin 30^\circ \\ &=\dfrac{1}{\sqrt{2}}\cdot\dfrac{\sqrt{3}}{2}-\dfrac{1}{\sqrt{2}}\cdot\dfrac{1}{2} \\ &=\dfrac{\sqrt{6}-\sqrt{2}}{4} \end{aligned}\)

For given ratios, use the quadrants to find the signs of any missing ratios before substitution.

In general, \(\sin(A+B)\ne\sin A+\sin B\).

Compound and Double-Angle Identities

Trigonometry

International Baccalaureate Mathematics: Applications and Interpretation Standard Level

The Addition Formulae

The Addition Formulae

Video Solution:

Solution:

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Split the angle	Working
\(105^\circ=60^\circ+45^\circ\)
\(75^\circ=45^\circ+30^\circ\)