Formula per sinusin dhe kosinusin | Shuma dhe diferenca e tyre

Formula për cos(x₁-x₂)

Marrim $\displaystyle \cos \left( 90{}^\text{o}-60{}^\text{o} \right)$ dhe $\displaystyle \cos 90{}^\text{o}-\cos 60{}^\text{o}$ dhe i krahasojmë. A janë të barabartë?

$\displaystyle \cos \left( 90{}^\text{o}-60{}^\text{o} \right)=\cos 30{}^\text{o}=\frac{\sqrt{2}}{2}$ , ndërsa $\displaystyle \cos 90{}^\text{o}-\cos 60{}^\text{o}=0-\frac{1}{2}=-\frac{1}{2}$ .

Pra, $\displaystyle \cos \left( 90{}^\text{o}-60{}^\text{o} \right)\ne \cos 90{}^\text{o}-\cos 60{}^\text{o}$ .

Nga shndërrime trigonometrike marrim formulën për diferencën e kosinuseve:

$\displaystyle \cos \left( {{x}_{1}}-{{x}_{2}} \right)=\cos {{x}_{1}}\cdot \cos {{x}_{2}}+\sin {{x}_{1}}\cdot \sin {{x}_{2}}$ .

Ushtrimi 1

Gjeni $\displaystyle \cos 120{}^\text{o}$

Zgjidhje

Për të gjetur $\displaystyle \cos 120{}^\text{o}$ , zbatojmë formulën e diferencës së kosinuseve:

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\displaystyle \cos \left( {{x}_{1}}-{{x}_{2}} \right)=\cos {{x}_{1}}\cdot \cos {{x}_{2}}+\sin {{x}_{1}}\cdot \sin {{x}_{2}}<pre class="ql-errors">*** QuickLaTeX cannot compile formula:
\[latex \displaystyle \cos 120{}^\text{o}$ e shkruajmë si $latex \displaystyle \cos \left( 180{}^\text{o}-60{}^\text{o} \right)$ dhe do të kemi:
$latex \displaystyle \cos \left( 180{}^\text{o}-60{}^\text{o} \right)=\cos 180{}^\text{o}\cdot \cos 60{}^\text{o}+\sin 180{}^\text{o}\cdot \sin 60{}^\text{o}$
$latex \displaystyle =-1\cdot \frac{1}{2}+0\cdot \frac{\sqrt{2}}{2}$
$latex \displaystyle =-1\cdot \frac{1}{2}+0\cdot \frac{\sqrt{2}}{2}=-\frac{1}{2}$.
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<h2 style="text-align: center;"><span style="color: #0000ff;"><strong>Formula për cos(x<sub>1</sub>+x<sub>2</sub>)</strong></span></h2>
Në identitetin që kemi për cos(x<sub>1</sub>+x<sub>2</sub>), zëvëndësojmë $latex \displaystyle {{x}_{2}}$ me $latex \displaystyle \left( -{{x}_{2}} \right)$.
Do të kemi:
<div style="overflow-x: auto;">$latex \displaystyle \cos \left( {{x}_{1}}+{{x}_{2}} \right)=\cos {{x}_{1}}\cdot \cos {{x}_{2}}-\sin {{x}_{1}}\cdot \sin {{x}_{2}}$, sepse $latex \displaystyle \cos \left( -{{x}_{2}} \right)=\cos \left( {{x}_{2}} \right)$ dhe $latex \displaystyle \sin \left( -{{x}_{2}} \right)=-\sin \left( {{x}_{2}} \right)$.</div>
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<h3 style="text-align: center;"><span style="text-decoration: underline; color: #ff0000;"><strong>Ushtrimi 2</strong></span></h3>
Të njehësohet $latex \displaystyle \cos 105{}^\text{o}$
<h4 style="text-align: center;"><span style="color: #008000;"><strong>Zgjidhje</strong></span></h4>
<div style="overflow-x: auto;">$latex \displaystyle \cos 105{}^\text{o}$ e shkruajmë si $latex \displaystyle \cos \left( 60{}^\text{o}+45{}^\text{o} \right)$ dhe zbatojmë formulën për gjetjen e shumës së kosinuseve:$latex \displaystyle \cos \left( {{x}_{1}}+{{x}_{2}} \right)=\cos {{x}_{1}}\cdot \cos {{x}_{2}}-\sin {{x}_{1}}\cdot \sin {{x}_{2}}$</div>
Bëjmë zëvëndësimet:
<div style="overflow-x: auto;">$latex \displaystyle \cos \left( 60{}^\text{o}+45{}^\text{o} \right)=\cos 60{}^\text{o}\cdot \cos 45{}^\text{o}-\sin 60{}^\text{o}\cdot \sin 45{}^\text{o}\]
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leading text: \[latex \displaystyle \cos 120{}^\text{o}$ 
</pre>latex \displaystyle =\frac{1}{2}\cdot \frac{\sqrt{2}}{2}-\frac{\sqrt{3}}{2}\cdot \frac{\sqrt{2}}{2}

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leading text: \displaystyle

$\displaystyle =\frac{\sqrt{2}}{4}-\frac{\sqrt{3}\cdot \sqrt{2}}{4}$

$\displaystyle =\frac{\sqrt{2}\left( 1-\sqrt{3} \right)}{4}$

Formula për sin(x₁+x₂)

Në identitetin për $\displaystyle \cos \left( {{x}_{1}}-{{x}_{2}} \right)=\cos {{x}_{1}}\cdot \cos {{x}_{2}}+\sin {{x}_{1}}\cdot \sin {{x}_{2}}$ , zëvëndësojmë $\displaystyle {{x}_{1}}$ me $\displaystyle \left( 90{}^\text{o}-{{x}_{1}} \right)$ dhe marrim:

*** QuickLaTeX cannot compile formula:
\displaystyle \cos \left( 90{}^\text{o}-{{x}_{1}}-{{x}_{2}} \right)=\cos \left( 90{}^\text{o}-{{x}_{1}} \right)\cdot \cos {{x}_{2}}+\sin \left( 90{}^\text{o}-{{x}_{1}} \right)\cdot \sin {{x}_{2}}<pre class="ql-errors">*** QuickLaTeX cannot compile formula:
\[latex \displaystyle \cos \left[ 90{}^\text{o}-\left( {{x}_{1}}-{{x}_{2}} \right) \right]=\sin {{x}_{1}}\cdot \cos {{x}_{2}}+\cos {{x}_{1}}\cdot \sin {{x}_{2}}$, pra:
$latex \displaystyle \sin \left( {{x}_{1}}+{{x}_{2}} \right)=\sin {{x}_{1}}\cdot \cos {{x}_{2}}+\cos {{x}_{1}}\cdot \sin {{x}_{2}}$.
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Kemi pasur parasysh faktin që $latex \displaystyle \sin \left( 90{}^\text{o}-\alpha  \right)=\cos \alpha $.
<h3 style="text-align: center;"><span style="text-decoration: underline; color: #ff0000;"><strong>Ushtrimi 3</strong></span></h3>
Të gjendet $latex \displaystyle \sin 75{}^\text{o}$.
<h4 style="text-align: center;"><span style="color: #008000;"><strong>Zgjidhje</strong></span></h4>
Shkruajmë $latex \displaystyle \sin 75{}^\text{o}=\sin \left( 45{}^\text{o}+30{}^\text{o} \right)$.
Zbatojmë formulën për shumën e sinuseve:
<div style="overflow-x: auto;">$latex \displaystyle \sin \left( 45{}^\text{o}+30{}^\text{o} \right)=\sin 45{}^\text{o}\cdot \cos 30{}^\text{o}+\cos 45{}^\text{o}\cdot \sin 30{}^\text{o}\]
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Display math should end with $$.
leading text: ...}_{2}}+\cos {{x}_{1}}\cdot \sin {{x}_{2}}$,
</pre>latex \displaystyle =\frac{\sqrt{2}}{2}\cdot \frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\cdot \frac{1}{2}

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leading text: \displaystyle

$\displaystyle =\frac{\sqrt{2}\left( \sqrt{3}+1 \right)}{4}$

Formula për sin(x₁-x₂)

Ka vend identiteti $\displaystyle \sin \left( {{x}_{1}}-{{x}_{2}} \right)=\sin {{x}_{1}}\cdot \cos {{x}_{2}}-\sin {{x}_{2}}\cdot \cos {{x}_{1}}$ .

Ushtrimi 4

Gjeni $\displaystyle \sin 15{}^\text{o}$ .

Zgjidhje

Shkruajmë $\displaystyle \sin 15{}^\text{o}=\sin \left( 45{}^\text{o}-30{}^\text{o} \right)$ .

Zbatojmë formulën për diferencën e sinuseve:

$\displaystyle \sin \left( {{x}_{1}}-{{x}_{2}} \right)=\sin {{x}_{1}}\cdot \cos {{x}_{2}}-\sin {{x}_{2}}\cdot \cos {{x}_{1}}$ $\displaystyle \sin \left( 45{}^\text{o}-30{}^\text{o} \right)=\sin 45{}^\text{o}\cdot \cos 30{}^\text{o}-\sin 30{}^\text{o}\cdot \cos 45{}^\text{o}$

$\displaystyle =\frac{\sqrt{2}}{2}\cdot \frac{\sqrt{3}}{2}-\frac{1}{2}\cdot \frac{\sqrt{2}}{2}$