Reciprocal ratios **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 These conversions require k 0. If cot x=0, solve cos x=0 directly, with sin x 0. Secant and cosecant cannot be zero. Quadratic in a ratio Factorise, solve each ratio equation, then list all angles in the 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 -1 <= sin x 1 -1 <= cos x 1 Reject sine or cosine values outside this range. Tangent can take any real value.

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

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

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

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 and quadratic trigonometric equations

Reciprocal and quadratic trigonometric equations