9.4: Jumla ya bidhaa na Bidhaa kwa-Jumla Formula

Malengo ya kujifunza

• Express bidhaa kama kiasi.
• Express kiasi kama bidhaa.

bendi maandamano chini ya shamba kujenga sauti ya ajabu kwamba bolsters umati. Sauti hiyo inasafiri kama wimbi linaloweza kutafsiriwa kwa kutumia kazi za trigonometric.

Kielelezo\(\PageIndex{1}\): Bendi ya maandamano ya UCLA (mikopo: Eric Chan, Flickr).

Kwa mfano, Kielelezo\(\PageIndex{2}\) inawakilisha wimbi sauti kwa muziki kumbuka A. katika sehemu hii, sisi kuchunguza utambulisho trigonometric kwamba ni msingi wa matukio ya kila siku kama vile mawimbi ya sauti.

Kielelezo\(\PageIndex{2}\)

Kuonyesha Bidhaa kama Sums

Tayari tumejifunza kanuni kadhaa muhimu kwa kupanua au kurahisisha maneno ya trigonometric, lakini wakati mwingine tunaweza kuhitaji kuelezea bidhaa za cosine na sine kama jumla. Tunaweza kutumia formula za bidhaa kwa jumla, ambazo zinaonyesha bidhaa za kazi za trigonometric kama jumla. Hebu tuchunguze utambulisho wa cosine kwanza na kisha utambulisho wa sine.

Kuonyesha Bidhaa kama Sums kwa Cosine

Tunaweza kupata formula ya bidhaa-kwa-jumla kutoka kwa utambulisho wa jumla na tofauti kwa cosine. Ikiwa tunaongeza equations mbili, tunapata:

\[\begin{align*} \cos \alpha \cos \beta+\sin \alpha \sin \beta&= \cos(\alpha-\beta)\\[4pt] \underline{+ \cos \alpha \cos \beta-\sin \alpha \sin \beta}&= \underline{ \cos(\alpha+\beta) }\\[4pt] 2 \cos \alpha \cos \beta&= \cos(\alpha-\beta)+\cos(\alpha+\beta)\end{align*}\]

Kisha, tunagawanya na 2 ili kutenganisha bidhaa za cosines:

\[ \cos \alpha \cos \beta= \dfrac{1}{2}[\cos(\alpha-\beta)+\cos(\alpha+\beta)] \label{eq1}\]

Jinsi ya: Kutokana na bidhaa ya cosines, onyesha kama jumla

1. Andika formula ya bidhaa za cosines.
2. Weka pembe zilizopewa katika formula.
3. Kurahisisha.

Mfano\(\PageIndex{1}\): Writing the Product as a Sum Using the Product-to-Sum Formula for Cosine

Andika bidhaa zifuatazo za cosines kama jumla:\(2\cos\left(\dfrac{7x}{2}\right) \cos\left(\dfrac{3x}{2}\right)\).

Suluhisho

Tunaanza kwa kuandika formula kwa bidhaa za cosines (Equation\ ref {eq1}):

\[ \cos \alpha \cos \beta = \dfrac{1}{2}[ \cos(\alpha-\beta)+\cos(\alpha+\beta) ] \nonumber \]

Tunaweza kisha kubadilisha pembe fulani katika formula na kurahisisha.

\[\begin{align*} 2 \cos\left(\dfrac{7x}{2}\right)\cos\left(\dfrac{3x}{2}\right)&= 2\left(\dfrac{1}{2}\right)[ \cos\left(\dfrac{7x}{2}-\dfrac{3x}{2}\right)+\cos\left(\dfrac{7x}{2}+\dfrac{3x}{2}\right) ]\\[4pt] &= \cos\left(\dfrac{4x}{2}\right)+\cos\left(\dfrac{10x}{2}\right) \\[4pt] &= \cos 2x+\cos 5x \end{align*}\]

Zoezi\(\PageIndex{1}\)

Tumia formula ya bidhaa-kwa-jumla (Equation\ ref {eq1}) kuandika bidhaa kama jumla au tofauti:\(\cos(2\theta)\cos(4\theta)\).

Jibu

\(\dfrac{1}{2}(\cos 6\theta+\cos 2\theta)\)

Kuonyesha Bidhaa ya Sine na Cosine kama Sum

Kisha, tutapata fomu ya bidhaa kwa jumla kwa sine na cosine kutoka kwa jumla na tofauti za sine. Ikiwa tunaongeza utambulisho wa jumla na tofauti, tunapata:

\[\begin{align*} \cos \alpha \cos \beta+\sin \alpha \sin \beta&= \cos(\alpha-\beta)\\[4pt] \underline{+ \cos \alpha \cos \beta-\sin \alpha \sin \beta}&= \cos(\alpha+\beta)\\[4pt] 2 \cos \alpha \cos \beta&= \cos(\alpha-\beta)+\cos(\alpha+\beta)\\[4pt] \text{Then, we divide by 2 to isolate the product of cosines:}\\[4pt] \cos \alpha \cos \beta&= \dfrac{1}{2}\left[\cos(\alpha-\beta)+\cos(\alpha+\beta)\right] \end{align*}\]

Mfano\(\PageIndex{2}\): Writing the Product as a Sum Containing only Sine or Cosine

Express bidhaa zifuatazo kama jumla zenye sine tu au cosine na hakuna bidhaa:\(\sin(4\theta)\cos(2\theta)\).

Suluhisho

Andika fomu ya bidhaa ya sine na cosine. Kisha ubadilisha maadili yaliyotolewa katika formula na ueleze.

\[\begin{align*} \sin \alpha \cos \beta&= \dfrac{1}{2}[ \sin(\alpha+\beta)+\sin(\alpha-\beta) ]\\[4pt] \sin(4\theta)\cos(2\theta)&= \dfrac{1}{2}[\sin(4\theta+2\theta)+\sin(4\theta-2\theta)]\\[4pt] &= \dfrac{1}{2}[\sin(6\theta)+\sin(2\theta)] \end{align*}\]

Zoezi\(\PageIndex{2}\)

Tumia formula ya bidhaa-kwa-jumla kuandika bidhaa kama jumla:\(\sin(x+y)\cos(x−y)\).

Jibu

\(\dfrac{1}{2}(\sin 2x+\sin 2y)\)

Kuonyesha Bidhaa za Sines katika Masharti ya Cosine

Kuelezea bidhaa za sines kwa suala la cosine pia linatokana na utambulisho wa jumla na tofauti kwa cosine. Katika kesi hii, tutaondoa kwanza kanuni mbili za cosine:

\[\begin{align*} \cos(\alpha-\beta)&= \cos \alpha \cos \beta+\sin \alpha \sin \beta\\[4pt] \underline{-\cos(\alpha+\beta)}&= -(\cos \alpha \cos \beta-\sin \alpha \sin \beta)\\[4pt] \cos(\alpha-\beta)-\cos(\alpha+\beta)&= 2 \sin \alpha \sin \beta\\[4pt] \text{Then, we divide by 2 to isolate the product of sines:}\\[4pt] \sin \alpha \sin \beta&= \dfrac{1}{2}[ \cos(\alpha-\beta)-\cos(\alpha+\beta) ] \end{align*}\]

Vile vile tunaweza kueleza bidhaa ya cosines katika suala la sine au hupata bidhaa nyingine kwa-jumla formula.

BIDHAA KWA JUMLA FORMULA

Fomu za bidhaa kwa jumla ni kama ifuatavyo:

\[\cos \alpha \cos \beta=\dfrac{1}{2}[\cos(\alpha−\beta)+\cos(\alpha+\beta)]\]

\[\sin \alpha \cos \beta=\dfrac{1}{2}[\sin(\alpha+\beta)+\sin(\alpha−\beta)]\]

\[\sin \alpha \sin \beta=\dfrac{1}{2}[\cos(\alpha−\beta)−\cos(\alpha+\beta)]\]

\[\cos \alpha \sin \beta=\dfrac{1}{2}[\sin(\alpha+\beta)−\sin(\alpha−\beta)]\]

Mfano\(\PageIndex{3}\): Express the Product as a Sum or Difference

Andika\(\cos(3\theta) \cos(5\theta)\) kama jumla au tofauti.

Suluhisho

Tuna bidhaa za cosines, kwa hiyo tunaanza kwa kuandika formula inayohusiana. Kisha sisi badala ya pembe zilizopewa na kurahisisha.

\[\begin{align*} \cos \alpha \cos \beta&= \dfrac{1}{2}[\cos(\alpha-\beta)+\cos(\alpha+\beta)]\\[4pt] \cos(3\theta)\cos(5\theta)&= \dfrac{1}{2}[\cos(3\theta-5\theta)+\cos(3\theta+5\theta)]\\[4pt] &= \dfrac{1}{2}[\cos(2\theta)+\cos(8\theta)]\qquad \text{Use even-odd identity} \end{align*}\]

Zoezi\(\PageIndex{3}\)

Tumia formula ya bidhaa-kwa-jumla ili kutathmini\(\cos \dfrac{11\pi}{12} \cos \dfrac{\pi}{12}\).

Jibu

\(\dfrac{−2−\sqrt{3}}{4}\)

Kuonyesha Sums kama Bidhaa

Baadhi ya matatizo yanahitaji reverse ya mchakato sisi tu kutumika. Fomu za jumla hadi bidhaa zinatuwezesha kuelezea kiasi cha sine au cosine kama bidhaa. Fomula hizi zinaweza inayotokana na utambulisho wa bidhaa-kwa-jumla. Kwa mfano, na mbadala chache, tunaweza hupata kiasi hadi bidhaa utambulisho kwa sine. Hebu\(\dfrac{u+v}{2}=\alpha\) na\(\dfrac{u−v}{2}=\beta\).

Kisha,

\[\begin{align*} \alpha+\beta&= \dfrac{u+v}{2}+\dfrac{u-v}{2}\\[4pt] &= \dfrac{2u}{2}\\[4pt] &= u \end{align*}\]

\[\begin{align*} \alpha-\beta&= \dfrac{u+v}{2}-\dfrac{u-v}{2}\\[4pt] &= \dfrac{2v}{2}\\[4pt] &= v \end{align*}\]

Hivyo, kuchukua nafasi\(\alpha\) na\(\beta\) katika formula ya bidhaa-kwa-jumla na maneno mbadala, tuna

\[\begin{align*} \sin \alpha \cos \beta&= \dfrac{1}{2}[\sin(\alpha+\beta)+\sin(\alpha-\beta)]\\[4pt] \sin \left ( \frac{u+v}{2} \right ) \cos \left ( \frac{u-v}{2} \right )&= \frac{1}{2}[\sin u + \sin v]\qquad \text{Substitute for } (\alpha+\beta) \text{ and } (\alpha\beta)\\[4pt] 2\sin\left(\dfrac{u+v}{2}\right) \cos\left(\dfrac{u-v}{2}\right)&= \sin u+\sin v \end{align*}\]

Utambulisho mwingine wa jumla hadi bidhaa hutolewa sawa.

JUMU-TO-BIDHAA FORMULA

Fomu za jumla kwa bidhaa ni kama ifuatavyo:

\[\sin \alpha+\sin \beta=2\sin\left(\dfrac{\alpha+\beta}{2}\right)\cos\left(\dfrac{\alpha−\beta}{2}\right)\]

\[\sin \alpha-\sin \beta=2\sin\left(\dfrac{\alpha-\beta}{2}\right)\cos\left(\dfrac{\alpha+\beta}{2}\right)\]

\[\cos \alpha−\cos \beta=−2\sin\left(\dfrac{\alpha+\beta}{2}\right)\sin\left(\dfrac{\alpha−\beta}{2}\right)\]

\[\cos \alpha+\cos \beta=2\sin\left(\dfrac{\alpha+\beta}{2}\right)\sin\left(\dfrac{\alpha−\beta}{2}\right)\]

Mfano\(\PageIndex{4}\): Writing the Difference of Sines as a Product

Andika tofauti zifuatazo za kujieleza sines kama bidhaa:\(\sin(4\theta)−\sin(2\theta)\).

Suluhisho

Tunaanza kwa kuandika formula kwa tofauti ya sines.

\[\begin{align*} \sin \alpha-\sin \beta&= 2\sin\left(\dfrac{\alpha-\beta}{2}\right)\cos\left(\dfrac{\alpha+\beta}{2}\right)\\[4pt] \text {Substitute the values into the formula, and simplify.}\\[4pt] \sin(4\theta)-\sin(2\theta)&= 2\sin\left(\dfrac{4\theta-2\theta}{2}\right) \cos\left(\dfrac{4\theta+2\theta}{2}\right)\\[4pt] &= 2\sin\left(\dfrac{2\theta}{2}\right) \cos\left(\dfrac{6\theta}{2}\right)\\[4pt] &= 2 \sin \theta \cos(3\theta) \end{align*}\]

Zoezi\(\PageIndex{4}\)

Tumia formula ya jumla hadi bidhaa ili kuandika jumla kama bidhaa:\(\sin(3\theta)+\sin(\theta)\).

Jibu

\(2\sin(2\theta)\cos(\theta)\)

Mfano\(\PageIndex{5}\): Evaluating Using the Sum-to-Product Formula

Tathmini\(\cos(15°)−\cos(75°)\). Angalia jibu kwa calculator ya graphing.

Suluhisho

Tunaanza kwa kuandika formula kwa tofauti ya cosines.

\\ kuanza {align*}
\ cos\ alpha-\ cos\ beta&= -2\ dhambi\ kushoto (\ dfrac {\ alpha+\ beta} {2}\ haki)\ dhambi\ kushoto (\ dfrac {\ alpha-\ beta} {2}\ haki)\\ [4pt]
\\ Nakala {Kisha sisi badala pembe zilizopewa na kurahisisha.}\
\ [4pt]\ 15^ {\ circ}) -\ cos (75^ {\ circ}) &= -2\ dhambi\ kushoto (\ dfrac {15^ {\ circ} +75^ {\ circ}} {2}\ haki)\ dhambi\ kushoto (\ dfrac {15^ {\ circ} -75^ {\ circ}} {2}\ haki)\\ [4pt]
&= -2\ dhambi (45^ {\ circ})\\ [4pt]
&= -2\ kushoto (\ dfrac {\ sqrt {2}} {2}\ kulia)\ kushoto (-\ drac {1} {2}\ haki)\\ [4pt]
&=\ dfrac {\ sqrt {2}} {2}
\ mwisho {align*}\]

Mfano\(\PageIndex{6}\): Proving an Identity

Thibitisha utambulisho:

\[\dfrac{\cos(4t)−\cos(2t)}{\sin(4t)+\sin(2t)}=−\tan t\]

Suluhisho

Tutaanza na upande wa kushoto, upande mgumu zaidi wa equation, na uandike tena maneno mpaka inafanana na upande wa kulia.

\[\begin{align*} \dfrac{\cos(4t)-\cos(2t)}{\sin(4t)+\sin(2t)}&= \dfrac{-2 \sin\left(\dfrac{4t+2t}{2}\right) \sin\left(\dfrac{4t-2t}{2}\right)}{2 \sin\left(\dfrac{4t+2t}{2}\right) \cos\left(\dfrac{4t-2t}{2}\right)}\\[4pt] &= \dfrac{-2 \sin(3t)\sin t}{2 \sin(3t)\cos t}\\[4pt] &= -\dfrac{\sin t}{\cos t}\\[4pt] &= -\tan t \end{align*}\]

Uchambuzi

Kumbuka kwamba kuthibitisha utambulisho wa trigonometric ina seti yake ya sheria. Taratibu za kutatua equation si sawa na taratibu za kuthibitisha utambulisho. Wakati sisi kuthibitisha utambulisho, sisi kuchukua upande mmoja kufanya kazi juu na kufanya mbadala mpaka upande huo ni kubadilishwa kuwa upande mwingine.

Mfano\(\PageIndex{7}\): Verifying the Identity Using Double-Angle Formulas and Reciprocal Identities

Thibitisha utambulisho\({\csc}^2 \theta−2=\cos(2\theta)\sin2\theta\).

Suluhisho

Kwa kuthibitisha equation hii, sisi ni kuleta pamoja kadhaa ya utambulisho. Tutatumia formula mbili-angle na utambulisho wa usawa. Tutafanya kazi na upande wa kulia wa equation na kuandika tena mpaka inafanana na upande wa kushoto.

\[\begin{align*} \cos(2\theta)\sin2\theta&= \dfrac{1-2 {\sin}^2 \theta}{{\sin}^2 \theta}\\[4pt] &= \dfrac{1}{{\sin}^2 \theta}-\dfrac{2 {\sin}^2 \theta}{{\sin}^2 \theta}\\[4pt] &= {\csc}^2 \theta - 2 \end{align*}\]

Zoezi\(\PageIndex{5}\)

Thibitisha utambulisho\(\tan \theta \cot \theta−{\cos}^2 \theta={\sin}^2 \theta\).

Jibu

\[\begin{align*} \tan \theta \cot \theta-{\cos}^2 \theta&= \left(\dfrac{\sin \theta}{\cos \theta}\right)\left(\dfrac{\cos \theta}{\sin \theta}\right)-{\cos}^2 \theta\\[4pt] &= 1-{\cos}^2 \theta\\[4pt] &= {\sin}^2 \theta \end{align*}\]

vyombo vya habari

Fikia rasilimali hizi za mtandaoni kwa maelekezo ya ziada na mazoezi na utambulisho wa bidhaa hadi jumla na jumla ya bidhaa.

• Jumla ya Utambulisho wa Bidhaa
• Jumla ya Bidhaa na Bidhaa kwa Utambulisho wa Jumla

Mlinganyo muhimu

Bidhaa kwa-jumla formula

\[\cos \alpha \cos \beta=\dfrac{1}{2}[\cos(\alpha−\beta)+\cos(\alpha+\beta)] \nonumber \]

\[\sin \alpha \cos \beta=\dfrac{1}{2}[\sin(\alpha+\beta)+\sin(\alpha−\beta)] \nonumber \]

\[\sin \alpha \sin \beta=\dfrac{1}{2}[\cos(\alpha−\beta)−\cos(\alpha+\beta)] \nonumber \]

\[\cos \alpha \sin \beta=\dfrac{1}{2}[\sin(\alpha+\beta)−\sin(\alpha−\beta)] \nonumber \]

Fomu za Jumla hadi bidhaa

\[\sin \alpha+\sin \beta=2\sin(\dfrac{\alpha+\beta}{2})\cos(\dfrac{\alpha−\beta}{2}) \nonumber \]

\[\sin \alpha-\sin \beta=2\sin(\dfrac{\alpha-\beta}{2})\cos(\dfrac{\alpha+\beta}{2}) \nonumber \]

\[\cos \alpha−\cos \beta=−2\sin(\dfrac{\alpha+\beta}{2})\sin(\dfrac{\alpha−\beta}{2}) \nonumber \]

\[\cos \alpha+\cos \beta=2\sin(\dfrac{\alpha+\beta}{2})\sin(\dfrac{\alpha−\beta}{2}) \nonumber \]

Dhana muhimu

• Kutoka kwa utambulisho wa jumla na tofauti, tunaweza kupata fomu za bidhaa hadi jumla na fomu za jumla kwa bidhaa kwa sine na cosine.
• Tunaweza kutumia fomu za bidhaa kwa jumla ili kuandika tena bidhaa za sines, bidhaa za cosines, na bidhaa za sine na cosine kama kiasi cha tofauti za sines na cosines. Angalia Mfano\(\PageIndex{1}\), Mfano\(\PageIndex{2}\), na Mfano\(\PageIndex{3}\).
• Tunaweza pia kupata utambulisho wa jumla hadi bidhaa kutoka kwa utambulisho wa bidhaa hadi jumla kwa kutumia mbadala.
• Tunaweza kutumia formula ya jumla hadi bidhaa ili kuandika upya jumla au tofauti ya sines, cosines, au bidhaa za sine na cosine kama bidhaa za sines na cosines. Angalia Mfano\(\PageIndex{4}\).
• Maneno ya trigonometric mara nyingi ni rahisi kutathmini kutumia formula. Angalia Mfano\(\PageIndex{5}\).
• Utambulisho unaweza kuthibitishwa kwa kutumia fomula nyingine au kwa kubadili maneno kuwa sines na cosines. Ili kuthibitisha utambulisho, tunachagua upande mgumu zaidi wa ishara sawa na kuandika tena mpaka itabadilishwa kuwa upande mwingine. Angalia Mfano\(\PageIndex{6}\) na Mfano\(\PageIndex{7}\).