የድምር ወይም የልዩነት ቀመሮች (Product-to-Sum and Sum-to-Product Identities)

አጭር ማጠቃለያ

ምርትን ወደ ድምር ወይም ልዩነት የሚቀይሩ ቀመሮች (Product-to-Sum Identities)

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

ድምርን ወይም ልዩነትን ወደ ምርት የሚቀይሩ ቀመሮች (Sum-to-Product Identities)

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

ቀመሮቹን ብቻ ሳይሆን የማውጣት ሂደታቸውንም አብሮ መማር ጥሩ ነው።

ቅድመ መስፈርቶች

• የትሪጎኖሜትሪ ፋንክሽኖች የመደመር ቀመሮች

ምርትን ወደ ድምር ወይም ልዩነት የሚቀይሩ ቀመሮች (Product-to-Sum Identities)

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

ማውጣት

የትሪጎኖሜትሪ ፋንክሽኖች የመደመር ቀመሮች

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

እንጠቀማለን።

($\ref{eqn:sin_add}$)+($\ref{eqn:sin_dif}$) ካደረግን

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

($\ref{eqn:sin_add}$)-($\ref{eqn:sin_dif}$) ካደረግን

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

በተመሳሳይ መንገድ

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

ከዚህ

($\ref{eqn:cos_add}$)+($\ref{eqn:cos_dif}$) ካደረግን

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

($\ref{eqn:cos_add}$)-($\ref{eqn:cos_dif}$) ካደረግን

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

ድምርን ወይም ልዩነትን ወደ ምርት የሚቀይሩ ቀመሮች (Sum-to-Product Identities)

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

ማውጣት

ምርትን ወደ ድምር ወይም ልዩነት የሚቀይሩ ቀመሮች (Product-to-Sum Identities) ከሚባሉት መነሳት፣ ድምርን ወይም ልዩነትን ወደ ምርት የሚቀይሩ ቀመሮች (Sum-to-Product Identities) እንዲሁም ማውጣት ይቻላል።

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

ብለን ካስቀመጥን እና ሁለቱን ስሌቶች በ $\alpha$, $\beta$ ላይ በአንድነት ካፈታን

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

ይህንንም ቀደም ሲል ባሉት ($\ref{sin_product_to_sum}$), ($\ref{cos_product_to_dif}$), ($\ref{cos_product_to_sum}$), ($\ref{sin_product_to_dif}$) ውስጥ በየተራ በመተካት የሚከተሉትን ቀመሮች እናገኛለን።

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