TRIGONOMETRY · HL
Sum and Product Formulas
Convert sums and differences to products — and back.
Section 1 of 3
Sums and differences to products
Suimeanna agus difríochtaí a thiontú ina n-iolraigh
$\sin A + \sin B$ ≠ $\sin (A + B)$
$\cos A + \cos B = 2\cos\dfrac{A+B}{2}\cos\dfrac{A-B}{2}$
$\cos A - \cos B = -2\sin\dfrac{A+B}{2}\sin\dfrac{A-B}{2}$
$\sin A + \sin B = 2\sin\dfrac{A+B}{2}\cos\dfrac{A-B}{2}$
$\sin A - \sin B = 2\cos\dfrac{A+B}{2}\sin\dfrac{A-B}{2}$
(i) Write $\cos 5A + \cos A$ as a product
$\cos A + \cos B = 2\cos\dfrac{A+B}{2}\cos\dfrac{A-B}{2}$ ✓
$= 2\cos 3A \cos 2A$
(ii) Write $\sin 4x + \sin x$ as a product
$\sin A + \sin B = 2\sin\dfrac{A+B}{2}\cos\dfrac{A-B}{2}$ ✓
$= 2\sin\dfrac{5x}{2}\cos\dfrac{3x}{2}$
Cannot mult by 2.
$\sin 5x \cos 3x$
Section 2 of 3
Show that the ratio equals $\cot 3A$
Show that $\dfrac{\cos 7A + \cos A}{\sin 7A - \sin A} = \cot 3A$.
$\dfrac{\cos 7A + \cos A}{\sin 7A - \sin A} = \dfrac{2\cos 4A \cos 3A}{2\cos 4A \sin 3A}$
$= \dfrac{\cos 3A}{\sin 3A}$
Section 3 of 3
Products to sums and differences
Iolraigh a thiontú ina suimeanna agus ina ndifríochtaí
$2\cos A\cos B = \cos(A + B) + \cos(A - B)$
$2\sin A\cos B = \sin(A + B) + \sin(A - B)$
$2\sin A\sin B = \cos(A - B) - \cos(A + B)$
$2\cos A\sin B = \sin(A + B) - \sin(A - B)$
(i) Change $2\sin 3A \sin A$ into a sum or difference
$2\sin A\sin B = \cos(A - B) - \cos(A + B)$
$2\sin 3A \sin A = \cos 2A - \cos 4A$
(ii) Change $\cos 3A \cos A$ to a sum or difference
$2\cos A\cos B = \cos(A + B) + \cos(A - B)$
$\cos A\cos B = \dfrac{1}{2}\left[\cos(A + B) + \cos(A - B)\right]$
$\cos 3A \cos A = \dfrac{1}{2}\left[\cos 4A + \cos 2A\right]$
SUM
The formulas in one box
Sums and differences to products
1.$\cos A + \cos B = 2\cos\dfrac{A+B}{2}\cos\dfrac{A-B}{2}$
2.$\cos A - \cos B = -2\sin\dfrac{A+B}{2}\sin\dfrac{A-B}{2}$
3.$\sin A + \sin B = 2\sin\dfrac{A+B}{2}\cos\dfrac{A-B}{2}$
4.$\sin A - \sin B = 2\cos\dfrac{A+B}{2}\sin\dfrac{A-B}{2}$
Products to sums and differences
5.$2\cos A\cos B = \cos(A + B) + \cos(A - B)$
6.$2\sin A\cos B = \sin(A + B) + \sin(A - B)$
7.$2\sin A\sin B = \cos(A - B) - \cos(A + B)$
8.$2\cos A\sin B = \sin(A + B) - \sin(A - B)$
End of lesson
Sum and Product Formulas — HL · Mathslive.ie