Chapter 3

Circular Measure and Trigonometry (Pure 1)

Radians and the arc and sector formulas they make possible, the exact values, the basic identities, and solving trigonometric equations across a stated interval.

หลักสูตรแกนกลาง / สสวท.: การวัดและเรขาคณิต (Measurement and Geometry)ตรงกับ ม.5 ฟังก์ชันตรีโกณมิติ (เพิ่มเติม) รวมเรเดียนและความยาวส่วนโค้ง
Common Core (US): F-TF, G-C
Cambridge: 9709 Pure Mathematics 1 — Circular measure; Trigonometry

3.1 Radians, arcs and sectors

The formulas, in radians only

π rad = 18 0^{\circ} s = r θ A = \frac{1}{2} r^{2} θ A_{segment} = \frac{1}{2} r^{2} (θ - sin θ)

A radian is defined as a ratio of two lengths, which is why these formulas have no $\frac{θ}{360}$ in them.

ข้อควรระวัง

From this chapter to the end of the book, radians are the unit. Every calculus result for $sin$ and $cos$ depends on it, so a calculator left in degree mode produces answers that are wrong in a way no later step will catch.

ตัวอย่าง

A sector of radius $12$ cm has perimeter $34$ cm. Find its angle in radians and its area.

วิธีทำ

1.Perimeter is the arc plus two radii: $s + 24 = 34$ , so $s = 10$
2. $θ = \frac{s}{r} = \frac{10}{12} = \frac{5}{6}$
3. $A = \frac{1}{2} (144) (\frac{5}{6})$

ตอบ $θ = \frac{5}{6}$ rad, area $60$ cm $^{2}$

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