Chapter 9

Further Trigonometry (Pure 3)

The addition and double-angle formulas, the reciprocal identities, and the R-formula — which rewrites $a sin θ + b cos θ$ as a single wave and thereby answers every question about its maximum.

หลักสูตรแกนกลาง / สสวท.: การวัดและเรขาคณิต (Measurement and Geometry)ตรงกับ ม.5 ฟังก์ชันตรีโกณมิติ (เพิ่มเติม) ส่วนเอกลักษณ์ผลบวกมุมและมุมสองเท่า
Common Core (US): F-TF
Cambridge: 9709 Pure Mathematics 3 — Trigonometry

9.1 Addition and double-angle formulas

Addition formulas

sin (A \pm B) = sin A cos B \pm cos A sin B cos (A \pm B) = cos A cos B \mp sin A sin B

The sign flips in the cosine version. That is not a typographical accident — it is the reason $cos 2 A$ has three different forms.

Double angle

sin 2 A = 2 sin A cos A cos 2 A = cos^{2} A - sin^{2} A = 2 cos^{2} A - 1 = 1 - 2 sin^{2} A

Three forms of $cos 2 A$ , and choosing the right one is most of the skill. Pick the one containing the ratio that already appears in the question.

The two rearrangements that integration needs

cos^{2} A = \frac{1}{2} (1 + cos 2 A) sin^{2} A = \frac{1}{2} (1 - cos 2 A)

These are the double-angle formulas turned inside out, and they are the only way to integrate a squared trigonometric function at this level.

ตัวอย่าง

Solve $cos 2 θ + 3 sin θ = 2$ for $0 \leq θ \leq 2 π$ .

วิธีทำ

1.The equation contains $sin θ$ , so use $cos 2 θ = 1 - 2 sin^{2} θ$
2. $1 - 2 sin^{2} θ + 3 sin θ = 2$
3. $2 sin^{2} θ - 3 sin θ + 1 = 0$ , so $(2 sin θ - 1) (sin θ - 1) = 0$
4. $sin θ = \frac{1}{2}$ or $sin θ = 1$

ตอบ $θ = \frac{π}{6}, \frac{π}{2}, \frac{5 π}{6}$

เคล็ดลับ

Choosing the form of $cos 2 A$ that matches the rest of the equation is the whole technique. Picking $2 cos^{2} θ - 1$ in that example would have left both $sin$ and $cos$ in the equation and nothing would factorise.

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