Some equations hide the basic ratio behind a reciprocal ratio or a square. Convert them into one or more basic equations first, then finish with the reference-angle method. **Reciprocal ratios: convert first** **Given** | **Convert to** | **Example** cosecx=k | sin x= 1k | cosecx=2 sin x= 12 sec x=k | cos x= 1k | sec x=- 2 cos x=- 1 2 cot x=k | tan x= 1k | cot x= 3 tan x= 1 3 **Quadratic in a ratio** Treat the ratio as the unknown of a quadratic: factorise, split into two basic equations, and solve each one over the stated interval. **Stage** | **Working** Factorise | 2sin2x+sin x-1=0 (2sin x-1)(sin x+1)=0 Two basic equations | sin x= 12 x=-1 Solve each for 0 <= x <= 360 | sin x= 12: x=30, 150 sin x=-1: x=270 All solutions | x=30, 150, 270 Sine and cosine only take values from -1 to 1, so reject a factor such as sin x=2 with a short note. Tangent can take any value — never reject a tangent value because of its size.

Some equations hide the basic ratio behind a reciprocal ratio or a square. Convert them into one or more basic equations first, then finish with the reference-angle method.

\(\operatorname{cosec}x=2\)
\(\sin x=\dfrac12\)

\(\sec x=-\sqrt2\)
\(\cos x=-\dfrac{1}{\sqrt2}\)

\(\cot x=\sqrt3\)
\(\tan x=\dfrac{1}{\sqrt3}\)

Treat the ratio as the unknown of a quadratic: factorise, split into two basic equations, and solve each one over the stated interval.

\(2\sin^2x+\sin x-1=0\)
\((2\sin x-1)(\sin x+1)=0\)

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

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

\(\sin x=\dfrac12:\ x=30^\circ,\ 150^\circ\)
\(\sin x=-1:\ x=270^\circ\)

Sine and cosine only take values from \(-1\) to \(1\), so reject a factor such as \(\sin x=2\) with a short note. Tangent can take any value — never reject a tangent value because of its size.

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

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