Chapter 7

Circular Measure

Radians — what they actually are, why the arc and sector formulas become one-liners in them, and why everything in the calculus chapters depends on using them.

หลักสูตรแกนกลาง / สสวท.: การวัดและเรขาคณิต (Measurement and Geometry)ตรงกับ ม.5 ฟังก์ชันตรีโกณมิติ (เพิ่มเติม) ส่วนเรเดียนและความยาวส่วนโค้ง
Common Core (US): F-TF, G-C
Cambridge: IGCSE 0606 — Topic 9: Circular measure

7.1 What a radian actually is

นิยาม

Radian

One radian is the angle at the centre of a circle subtended by an arc equal in length to the radius. It is a ratio of two lengths, so it has no units — and that is the reason the formulas below have no stray fractions in them.

Converting

π rad = 18 0^{\circ} 1 rad \approx 57. 3^{\circ}

Leave answers as exact multiples of $π$ unless the question asks for a decimal. $\frac{π}{6}$ is a better answer than $0.524$ .

ข้อควรระวัง

This is the chapter that decides whether the last three chapters of this book work. Every calculus result about $sin$ and $cos$ — including $\frac{d}{d x} sin x = cos x$ — is true ONLY in radians. A calculator left in degree mode gives answers that are wrong in a way nothing later will flag.

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The whole chapter is written; we read it back section by section before releasing it, because maths nobody has checked should not be on a public page. What is open is complete — nothing here is abridged.

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