Trigonometry

The three reciprocal ratios, the identities that connect everything, solving trigonometric equations across a stated range, and reading amplitude and period off a graph.

หลักสูตรแกนกลาง / สสวท.: การวัดและเรขาคณิต (Measurement and Geometry)ต่อยอดจาก ม.4 อัตราส่วนตรีโกณมิติ ไปสู่ ม.5 ฟังก์ชันตรีโกณมิติ (เพิ่มเติม)
Common Core (US): F-TF, G-SRT
Cambridge: IGCSE 0606 — Topic 10: Trigonometry

8.1 Reciprocal ratios and exact values

The three new ratios

csc θ = \frac{1}{sin θ} sec θ = \frac{1}{cos θ} cot θ = \frac{1}{tan θ} = \frac{cos θ}{sin θ}

The pairing is deliberately awkward: secant goes with cosine, cosecant with sine. Check the THIRD letter of the long name and it comes out right every time.

$θ$	$sin θ$	$cos θ$	$tan θ$
$0$	$0$	$1$	$0$
$\frac{π}{6}$	$\frac{1}{2}$	$\frac{3}{2}$	$\frac{1}{3}$
$\frac{π}{4}$	$\frac{1}{2}$	$\frac{1}{2}$	$1$
$\frac{π}{3}$	$\frac{3}{2}$	$\frac{1}{2}$	$3$
$\frac{π}{2}$	$1$	$0$	undefined

The exact values worth knowing by heart

เคล็ดลับ

These come from two triangles, not from memory: half an equilateral triangle of side $2$ , and a right-angled isosceles triangle with two sides of $1$ . Redrawing them takes ten seconds and cannot be misremembered under pressure.

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