<p class="paragraph-header">Reciprocal ratios</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 540px;"><colgroup><col style="width: 29.6296%;"><col style="width: 31.4815%;"><col style="width: 38.8889%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Given</strong></th><th style="text-align: center; --cell-justify: center"><strong>Convert to</strong></th><th style="text-align: center; --cell-justify: center"><strong>Example</strong></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">\(\operatorname{cosec}x=k\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></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">\(\sec x=k\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></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">\(\cot x=k\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr></tbody></table></div></div>
<p>These conversions require \(k\ne0\). If&nbsp;\(\cot x=0\), solve \(\cos x=0\) directly, with \(\sin x\ne0\). Secant and cosecant cannot be zero.</p>
<p class="paragraph-header">Quadratic in a ratio</p>
<p>Factorise, solve each ratio equation, then list all angles in the interval.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 540px;"><colgroup><col style="width: 40.7407%;"><col style="width: 59.2593%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Stage</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">Factorise</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">Two basic equations</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">Solve each for&nbsp;\(0^\circ\le x\le 360^\circ\)</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">All solutions</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr></tbody></table></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">\(-1\le\sin x\le1\qquad-1\le\cos x\le1\)</p></div></div>
<p>Reject sine or cosine values outside this range. Tangent can take any real value.</p>

\(\begin{aligned} \operatorname{cosec}x&=2 \\ \sin x&=\dfrac12 \end{aligned}\)

\(\begin{aligned} \sec x&=-\sqrt2 \\ \cos x&=-\dfrac{1}{\sqrt2} \end{aligned}\)

\(\begin{aligned} \cot x&=\sqrt3 \\ \tan x&=\dfrac{1}{\sqrt3} \end{aligned}\)

These conversions require \(k\ne0\). If \(\cot x=0\), solve \(\cos x=0\) directly, with \(\sin x\ne0\). Secant and cosecant cannot be zero.

Factorise, solve each ratio equation, then list all angles in the interval.

\(\begin{aligned} 2\sin^2x+\sin x-1&=0 \\ (2\sin x-1)(\sin x+1)&=0 \end{aligned}\)

\(\sin x=\dfrac12\quad\text{or}\quad\sin x=-1\)

Solve each for \(0^\circ\le x\le 360^\circ\)

\(\begin{aligned} \sin x&=\dfrac12:\ x=30^\circ,\ 150^\circ \\ \sin x&=-1:\ x=270^\circ \end{aligned}\)

Reject sine or cosine values outside this range. Tangent can take any real value.

Reciprocal-Ratio and Quadratic Equations

Reciprocal-Ratio and Quadratic Equations

Stage	Working
Factorise
Two basic equations
Solve each for \(0^\circ\le x\le 360^\circ\)
All solutions

Given	Convert to	Example
\(\operatorname{cosec}x=k\)
\(\sec x=k\)
\(\cot x=k\)