Kanuni za kubadili zao kuwa jumla au tofauti na jumla au tofauti kuwa zao (Product-to-Sum and Sum-to-Product Identities)

TL;DR

Kanuni za kubadili zao kuwa jumla au tofauti (Product-to-Sum Identities)

• \[\sin \alpha \cos \beta = \frac { 1 } { 2 } \{ \sin ( \alpha + \beta ) + \sin ( \alpha - \beta ) \}\]
• \[\cos \alpha \sin \beta = \frac { 1 } { 2 } \{ \sin ( \alpha + \beta ) - \sin ( \alpha - \beta ) \}\]
• \[\cos \alpha \cos \beta = \frac { 1 } { 2 } \{ \cos ( \alpha + \beta ) + \cos ( \alpha - \beta )\}\]
• \[\sin \alpha \sin \beta = - \frac { 1 } { 2 } \{ \cos ( \alpha + \beta ) - \cos ( \alpha - \beta ) \}\]

Kanuni za kubadili jumla au tofauti kuwa zao (Sum-to-Product Identities)

• \[\sin A + \sin B = 2\sin \frac{A+B}{2}\cos \frac{A-B}{2}\]
• \[\sin A - \sin B = 2\cos \frac{A+B}{2}\sin \frac{A-B}{2}\]
• \[\cos A + \cos B = 2\cos \frac{A+B}{2}\cos \frac{A-B}{2}\]
• \[\cos A - \cos B = -2\sin \frac{A+B}{2}\sin \frac{A-B}{2}\]

Ni vyema kujifunza si kanuni pekee, bali pia mchakato wa utoaji wake.

Mahitaji ya awali

• Kanuni za kuongeza za trigonometria

Kanuni za kubadili zao kuwa jumla au tofauti (Product-to-Sum Identities)

• \[\sin \alpha \cos \beta = \frac { 1 } { 2 } \{ \sin ( \alpha + \beta ) + \sin ( \alpha - \beta ) \}\]
• \[\cos \alpha \sin \beta = \frac { 1 } { 2 } \{ \sin ( \alpha + \beta ) - \sin ( \alpha - \beta ) \}\]
• \[\cos \alpha \cos \beta = \frac { 1 } { 2 } \{ \cos ( \alpha + \beta ) + \cos ( \alpha - \beta )\}\]
• \[\sin \alpha \sin \beta = - \frac { 1 } { 2 } \{ \cos ( \alpha + \beta ) - \cos ( \alpha - \beta ) \}\]

Utoaji

Kwa kutumia kanuni za kuongeza za trigonometria,

\[\begin{align} \sin(\alpha+\beta) &= \sin \alpha \cos \beta + \cos \alpha \sin \beta \tag{1}\label{eqn:sin_add}\\ \sin(\alpha-\beta) &= \sin \alpha \cos \beta - \cos \alpha \sin \beta \tag{2}\label{eqn:sin_dif} \end{align}\]

Tukijumlisha ($\ref{eqn:sin_add}$)+($\ref{eqn:sin_dif}$), tunapata

\[\sin(\alpha+\beta) + \sin(\alpha-\beta) = 2 \sin \alpha \cos \beta \tag{3}\label{sin_product_to_sum}\] \[\therefore \sin \alpha \cos \beta = \frac { 1 } { 2 } \{ \sin ( \alpha + \beta ) + \sin ( \alpha - \beta ) \}.\]

Tukitoa ($\ref{eqn:sin_add}$)-($\ref{eqn:sin_dif}$), tunapata

\[\sin(\alpha+\beta) - \sin(\alpha-\beta) = 2 \cos \alpha \sin \beta \tag{4}\label{cos_product_to_dif}\] \[\therefore \cos \alpha \sin \beta = \frac { 1 } { 2 } \{ \sin ( \alpha + \beta ) - \sin ( \alpha - \beta ) \}.\]

Kwa njia hiyo hiyo,

\[\begin{align} \cos(\alpha+\beta) &= \cos \alpha \cos \beta - \sin \alpha \sin \beta \tag{5}\label{eqn:cos_add} \\ \cos(\alpha-\beta ) &= \cos \alpha \cos \beta + \sin \alpha \sin \beta \tag{6}\label{eqn:cos_dif} \end{align}\]

kutoka hapa,

Tukijumlisha ($\ref{eqn:cos_add}$)+($\ref{eqn:cos_dif}$), tunapata

\[\cos(\alpha+\beta) + \cos(\alpha-\beta) = 2 \cos \alpha \cos \beta \tag{7}\label{cos_product_to_sum}\] \[\therefore \cos \alpha \cos \beta = \frac { 1 } { 2 } \{ \cos(\alpha+\beta) + \cos(\alpha-\beta) \}.\]

Tukitoa ($\ref{eqn:cos_add}$)-($\ref{eqn:cos_dif}$), tunapata

\[\cos(\alpha+\beta) - \cos(\alpha-\beta) = -2 \sin \alpha \sin \beta \tag{8}\label{sin_product_to_dif}\] \[\therefore \sin \alpha \sin \beta = -\frac { 1 } { 2 } \{ \cos(\alpha+\beta) - \cos(\alpha-\beta) \}.\]

Kanuni za kubadili jumla au tofauti kuwa zao (Sum-to-Product Identities)

• \[\sin A + \sin B = 2\sin \frac{A+B}{2}\cos \frac{A-B}{2}\]
• \[\sin A - \sin B = 2\cos \frac{A+B}{2}\sin \frac{A-B}{2}\]
• \[\cos A + \cos B = 2\cos \frac{A+B}{2}\cos \frac{A-B}{2}\]
• \[\cos A - \cos B = -2\sin \frac{A+B}{2}\sin \frac{A-B}{2}\]

Utoaji

Kutoka kwa kanuni za kubadili zao kuwa jumla au tofauti (Product-to-Sum Identities), tunaweza pia kutoa kanuni za kubadili jumla au tofauti kuwa zao (Sum-to-Product Identities).

\[\alpha + \beta = A, \quad \alpha - \beta = B\]

Tukiweka hivyo na kutatua mfumo wa milinganyo hiyo miwili kwa $\alpha$ na $\beta$, tunapata

\[\alpha = \frac{A+B}{2}, \quad \beta = \frac{A-B}{2}.\]

Tukibadilisha haya katika ($\ref{sin_product_to_sum}$), ($\ref{cos_product_to_dif}$), ($\ref{cos_product_to_sum}$), na ($\ref{sin_product_to_dif}$) mtawalia, tunapata kanuni zifuatazo.

\[\begin{align*} \sin A + \sin B &= 2\sin \frac{A+B}{2}\cos \frac{A-B}{2} \\ \sin A - \sin B &= 2\cos \frac{A+B}{2}\sin \frac{A-B}{2} \\ \cos A + \cos B &= 2\cos \frac{A+B}{2}\cos \frac{A-B}{2} \\ \cos A - \cos B &= -2\sin \frac{A+B}{2}\sin \frac{A-B}{2}. \end{align*}\]