Compound angle formula and Multiple angle formula

Addition Formulae and Double Angle Formulae

6.5

Addition Formulae and Double Angle Formulae

The addition formulae that are used to find trigonometry ratios of addition angles are as follows:


	\(\text{sin}(A+B)= \text{sin} \ A \text{ cos} \ B + \text{cos} \ A \text{ sin} \ B\) \(\text{sin}(A-B)= \text{sin} \ A \text{ cos} \ B - \text{cos} \ A \text{ sin} \ B\)

	\(\text{cos}(A+B)= \text{cos} \ A \text{ cos} \ B - \text{sin} \ A \text{ sin} \ B\) \(\text{cos}(A-B)= \text{cos} \ A \text{ cos} \ B + \text{sin} \ A \text{ sin} \ B\)

	\(\text{tan}(A+B) = \dfrac{\text{tan }A + \text{tan }B}{1-\text{tan }A \text{ tan }B}\) \(\text{tan}(AB) = \dfrac{\text{tan }A - \text{tan }B}{1+\text{tan }A \text{ tan }B}\)

Angles in the form \((A+B) \text{ or }(A-B)\) are called addition angles
Angles in the form \(2A, \ 3A, \ ...\) are known as double angles

The same method can be used to verify the other addition formulae

Example

(a)

Prove the identities for

\(\text{sin }(90^{\circ}+A) = \text{cos } A\).

(b)

Without using a calculator, find the value of \(\text{tan }15^{\circ}\).

Solution:

(a)

\(\text{sin }(90^{\circ}+A) \\ = \text{sin }90^{\circ} \text{ cos }A + \text{ cos }90^{\circ} \text{ sin }A\\ =(1) \text{ cos }A + (0) \text{ sin }A\\ =\text{ cos }A \)

(b)

Recall that:

Angles	sin	cos	tan
\(45^{\circ}\)	\(\dfrac{1}{\sqrt2}\)	\(\dfrac{1}{\sqrt2}\)	\(1\)
\(60^{\circ}\)	\(\dfrac{\sqrt3}{2}\)	\(\dfrac{1}{2}\)	\(\sqrt3\)

\(\begin{aligned} &\text{tan }15^{\circ}\\ &=\text{tan }(60^{\circ} -45^{\circ})\\\\ &=\dfrac{\text{tan }60^{\circ} - \text{tan }45^{\circ}}{1+\text{tan }60^{\circ} \ \text{tan }45^{\circ}}\\\\ &=\dfrac{\sqrt 3 - 1}{1+ (\sqrt3)(1)}\\\\ &=\dfrac{\sqrt3 -1}{\sqrt 3+1}\\\\ &=2-\sqrt3 \end{aligned} \)

The double-angle formulae:


	\(\text{sin } 2A = 2 \text{ sin }A\text{ cos }A\)

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

	\(\text{tan }2A = \dfrac{2\text{ tan }A}{1-\text{tan}^2 \ A}\)

Other formulae involving double angles can be derived by induction
This relation can be used to prove half-angle formulae where \(\text{sin }\dfrac{A}{2}\) , \(\text{kos }\dfrac{A}{2}\) and \(\text{tan }\dfrac{A}{2}\) are expressed in terms of \(\text{sin }A\) dan \(\text{cos }A\) as below:


	\(\text{sin }\dfrac{A}{2} = \pm \sqrt{\dfrac{1-\text{ cos }A}{2}}\)

	\(\text{cos }\dfrac{A}{2} = \pm \sqrt{\dfrac{1+\text{ cos }A}{2}}\)

	\(\text{tan }\dfrac{A}{2} = \pm \sqrt{\dfrac{\text{sin }A}{1+\text{ cos }A}}\)

Example

(a)	Find the value for the expression \(2\text{ sin }15^{\circ} \text{ cos }15^{\circ}\) using the double-angle formulae.

(b)	Prove that \(\text{tan }\dfrac{x}{2} = \dfrac{1-\text{ cos }x}{\text{sin }x}\).


Penyelesaian:

(a)	\(2\text{ sin }15^{\circ} \text{ cos }15^{\circ}\\ = \text{sin }2(15^{\circ})\\ =\text{sin }30^{\circ}\\ =\dfrac{1}{2}\)

(b)	\(\begin{aligned} \text{Sebelah kanan } &= \dfrac{1-\text{ cos }x}{\text{sin }x}\\\\ &= \dfrac{1-\begin{pmatrix} 1-2\text{ sin}^2 \ \dfrac{x}{2} \end{pmatrix}}{2 \text{ sin }\dfrac{x}{2}\text{ cos }\dfrac{x}{2}}\\\\ &= \dfrac{2\text{ sin}^2 \ \dfrac{x}{2} }{2 \text{ sin }\dfrac{x}{2}\text{ cos }\dfrac{x}{2}}\\\\ &=\dfrac{\text{sin }\dfrac{x}{2}}{\text{cos }\dfrac{x}{2}}\\\\ &=\text{tan }\dfrac{x}{2} \end{aligned}\) Hence, it is proven that \(\text{tan }\dfrac{x}{2} = \dfrac{1-\text{ cos }x}{\text{sin }x}\).

Addition Formulae and Double Angle Formulae

Chapter : Trigonometric Functions

Topic : Compound angle formula and Multiple angle formula

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