Goniometrická rovnice

Goniometrická rovnice je tehdy, pokud je neznámá v goniometrické funkci. ^[1] K vyřešení goniometrické rovnice se používá jednotková kružnice.

Příklad, jak může goniometrická rovnice vypadat:

${\displaystyle (\sin x)^{2}+2\sin x-3=0}$

Řešení goniometrické rovnice

^{[2] [3]}

Jednoduché rovnice

1. rovnice

1. ${\displaystyle \cos x=-{\frac {\sqrt {3}}{2}}}$
2. ${\displaystyle x_{1}={\frac {5\pi }{6}}+2k\pi ,k\in \mathbb {Z} }$
3. ${\displaystyle x_{2}={\frac {7\pi }{6}}+2k\pi ,k\in \mathbb {Z} }$

2. rovnice

1. ${\displaystyle {\textrm {tg}}\,x=-{\sqrt {3}}}$
2. ${\displaystyle x={\frac {2\pi }{3}}+k\pi ,k\in \mathbb {Z} }$

Substituce

1. rovnice

1. ${\displaystyle (\sin x)^{2}+2\sin x-3=0}$
2. Zavedeme substituci ${\displaystyle a=\sin x}$:
   ${\displaystyle a^{2}+2a-3=0}$
3. Vypočítáme kvadratickou rovnici:
   ${\displaystyle a_{1,2}={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}={\frac {-2\pm {\sqrt {2^{2}-4\cdot 1\cdot (-3)}}}{2\cdot 1}}={\frac {-2\pm {\sqrt {16}}}{2}}={\frac {-2\pm 4}{2}}}$
   
   ${\displaystyle a_{1}={\frac {-2+4}{2}}={\frac {2}{2}}=1}$
   
   ${\displaystyle a_{2}={\frac {-2-4}{2}}={\frac {-6}{2}}=-3}$
4. Nyní si můžeme napsat 2 rovnice:
   1. ${\displaystyle \sin x=1}$
   2. ${\displaystyle \sin x=-3}$
5. Vyřešíme obě rovnice:
   1. ${\displaystyle \sin x=1}$
      ${\displaystyle x={\frac {1}{2}}\pi +2k\pi }$
   2. ${\displaystyle \sin x=-3}$
      ${\displaystyle x=\phi }$

Tím je vyřešená goniometrická rovnice pomocí substituce.

2. rovnice

1. ${\displaystyle \sin \left(x+{\frac {\pi }{6}}\right)=1}$
2. Zavedeme substituci ${\displaystyle a=x+{\frac {\pi }{6}}}$:
   ${\displaystyle \sin a=1}$
3. ${\displaystyle a={\frac {\pi }{2}}+2k\pi }$
4. Dosadíme substituci ${\displaystyle a=x+{\frac {\pi }{6}}}$:
   ${\displaystyle x+{\frac {\pi }{6}}={\frac {\pi }{2}}+2k\pi }$
5. ${\displaystyle a=x+{\frac {\pi }{6}}}$:
   ${\displaystyle x={\frac {3\pi }{6}}+2k\pi -{\frac {\pi }{6}}}$
6. ${\displaystyle x={\frac {2\pi }{6}}+2k\pi }$
7. ${\displaystyle x={\frac {\pi }{3}}+2k\pi }$

Tím je vyřešená goniometrická rovnice pomocí substituce.

Rovnice s více funkcemi současně

1. rovnice

1. ${\displaystyle {\sqrt {3}}\cos x=2-\sin x}$

2. umocníme rovnici na druhou:

${\displaystyle 3\cos ^{2}x=(2-\sin x)^{2}}$

3. použijeme vzorec ${\displaystyle \cos ^{2}x=1-\sin ^{2}x}$

${\displaystyle 3-3\sin ^{2}x=4-4\sin x+\sin ^{2}x}$

4. ${\displaystyle 0=4\sin ^{2}x-4\sin x+1}$

5. použijeme vzorec ${\displaystyle a^{2}-2ab+b^{2}=(a-b)^{2}}$

${\displaystyle (2\sin x-1)^{2}=0}$

6. celou rovnici odmocníme:

${\displaystyle 2\sin x-1=0}$

7. ${\displaystyle \sin x={\frac {1}{2}}}$

${\displaystyle x_{\scriptstyle {\text{1}}}={\frac {\pi }{6}}+2k\pi }$

${\displaystyle x_{\scriptstyle {\text{2}}}={\frac {5\pi }{6}}+2k\pi }$

8. z důvodu neekvivalentních úprav 2. a 6. je nutná zkouška

kořen ${\displaystyle x_{\scriptstyle {\text{2}}}}$ rovnici nevyhovuje a jediným řešením je ${\displaystyle x_{\scriptstyle {\text{1}}}}$

Takto je možné řešit rovnice se dvěma různými goniometrickými funkcemi

2. rovnice

1. ${\displaystyle (\cot x)^{-1}=-(\tan x)^{-1}+2(\sin x)^{-1}}$
2. Použijeme vztahy mezi funkcemi:

${\displaystyle \tan x=2(\sin x)^{-1}-\cot x}$

${\displaystyle {\frac {\sin x}{\cos x}}={\frac {2}{\sin x}}-{\frac {\cos x}{\sin x}}}$

1. zbavíme se zlomků:

${\displaystyle \sin ^{2}x=\cos x*(2-\cos x)}$

1. Použijeme vzorec ${\displaystyle \sin ^{2}x=1-\cos ^{2}x}$

${\displaystyle 1-\cos ^{2}x=2\cos x-\cos ^{2}x}$

1. ${\displaystyle 1=2\cos x}$
2. ${\displaystyle \cos x=1/2}$
3. ${\displaystyle x_{\scriptstyle {\text{1}}}={\frac {\pi }{6}}+2k\pi }$

${\displaystyle x_{\scriptstyle {\text{2}}}={\frac {11\pi }{6}}+2k\pi }$

1. Rovnice vyřešena

Vybrané (nejpoužívanější) vzorce

Podrobnější informace naleznete v článku Goniometrická funkce #Vybrané vzorce z oblasti goniometrie.

^{[4] [5]}

• Záporné hodnoty úhlů
  • ${\displaystyle \sin(-\alpha )=-\sin \alpha \,\!}$
  • ${\displaystyle \cos(-\alpha )=\cos \alpha \,\!}$
  • ${\displaystyle \mathrm {tg} (-\alpha )=-\mathrm {tg} \,\alpha \,\!}$
  • ${\displaystyle \mathrm {cotg} (-\alpha )=-\mathrm {cotg} \,\alpha \,\!}$
• Vzájemné vztahy mezi goniometrickými funkcemi stejného úhlu
  • ${\displaystyle \sin ^{2}\alpha +\cos ^{2}\alpha =1\,\!}$
  • ${\displaystyle \mathrm {tg} \,\alpha \cdot \mathrm {cotg} \,\alpha =1\,\!}$
  • ${\displaystyle {\textrm {tg}}\,\alpha ={\frac {\sin \alpha }{\cos \alpha }}\,\!}$
  • ${\displaystyle {\textrm {cotg}}\,\alpha ={\frac {\cos \alpha }{\sin \alpha }}\,\!}$
  • ${\displaystyle \sin \alpha ={\sqrt {1-\cos ^{2}\alpha }}}$
  • ${\displaystyle \cos \alpha ={\sqrt {1-\sin ^{2}\alpha }}}$
  • ${\displaystyle {\textrm {tg}}\,\alpha ={\frac {1}{{\textrm {cotg}}\,\alpha }}\,\!}$
• Dvojnásobný úhel
  • ${\displaystyle \sin 2\alpha =2\cdot \sin \alpha \cos \alpha \,\!}$
  • ${\displaystyle \cos 2\alpha =\cos ^{2}\alpha -\sin ^{2}\alpha \,\!}$
• Poloviční úhel
  • ${\displaystyle \sin {\frac {\alpha }{2}}={\sqrt {\frac {1-\cos \alpha }{2}}}\,\!}$
  • ${\displaystyle \cos {\frac {\alpha }{2}}={\sqrt {\frac {1+\cos \alpha }{2}}}\,\!}$
• Mocniny goniometrických funkcí
  • ${\displaystyle \sin ^{2}\alpha ={\frac {1}{2}}(1-\cos 2\alpha )}$
  • ${\displaystyle \cos ^{2}\alpha ={\frac {1}{2}}(1+\cos 2\alpha )}$
• Goniometrické funkce součtu a rozdílu úhlů
  • ${\displaystyle \sin \left(\alpha \pm \beta \right)=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta \,\!}$
  • ${\displaystyle \cos \left(\alpha \pm \beta \right)=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta \,\!}$

Kvadranty a hodnoty funkcí ve vybraných úhlech

Jednotková kružnice

^[6]

Kvadrant	α	sin α	cos α	tg α	cotg α
1. kvadrant	0° – 90°	+	+	+	+
2. kvadrant	90° – 180°	+	–	–	-
3. kvadrant	180° – 270°	–	–	+	+
4. kvadrant	270° – 360°	–	+	–	-

Stupně	Radiány	Sinus	Kosinus	Tangens	Kotangens
0	${\displaystyle 0\,}$	${\displaystyle 0\,}$	${\displaystyle 1\,}$	${\displaystyle 0\,}$	${\displaystyle -\,}$
30	${\displaystyle {\frac {\pi }{6}}}$	${\displaystyle {\frac {1}{2}}}$	${\displaystyle {\frac {\sqrt {3}}{2}}}$	${\displaystyle {\frac {\sqrt {3}}{3}}}$	${\displaystyle {\sqrt {3}}}$
45	${\displaystyle {\frac {\pi }{4}}}$	${\displaystyle {\frac {\sqrt {2}}{2}}}$	${\displaystyle {\frac {\sqrt {2}}{2}}}$	${\displaystyle 1\,}$	${\displaystyle 1\,}$
60	${\displaystyle {\frac {\pi }{3}}}$	${\displaystyle {\frac {\sqrt {3}}{2}}}$	${\displaystyle {\frac {1}{2}}}$	${\displaystyle {\sqrt {3}}}$	${\displaystyle {\frac {\sqrt {3}}{3}}}$
90	${\displaystyle {\frac {\pi }{2}}}$	${\displaystyle 1\,}$	${\displaystyle 0\,}$	${\displaystyle -\,}$	${\displaystyle 0\,}$
120	${\displaystyle {\frac {2\pi }{3}}}$	${\displaystyle {\frac {\sqrt {3}}{2}}}$	${\displaystyle -{\frac {1}{2}}}$	${\displaystyle -{\sqrt {3}}}$	${\displaystyle -{\frac {\sqrt {3}}{3}}}$
135	${\displaystyle {\frac {3\pi }{4}}}$	${\displaystyle {\frac {\sqrt {2}}{2}}}$	${\displaystyle -{\frac {\sqrt {2}}{2}}}$	${\displaystyle -1\,}$	${\displaystyle -1\,}$
150	${\displaystyle {\frac {5\pi }{6}}}$	${\displaystyle {\frac {1}{2}}}$	${\displaystyle {\frac {-{\sqrt {3}}}{2}}}$	${\displaystyle {\frac {-{\sqrt {3}}}{3}}}$	${\displaystyle -{\sqrt {3}}}$
180	${\displaystyle \pi \,}$	${\displaystyle 0\,}$	${\displaystyle -1\,}$	${\displaystyle 0\,}$	${\displaystyle -\,}$
210	${\displaystyle {\frac {7\pi }{6}}}$	${\displaystyle -{\frac {1}{2}}}$	${\displaystyle -{\frac {\sqrt {3}}{2}}}$	${\displaystyle {\frac {\sqrt {3}}{3}}}$	${\displaystyle {\sqrt {3}}}$
225	${\displaystyle {\frac {5\pi }{4}}}$	${\displaystyle -{\frac {\sqrt {2}}{2}}}$	${\displaystyle -{\frac {\sqrt {2}}{2}}}$	${\displaystyle 1\,}$	${\displaystyle 1\,}$
240	${\displaystyle {\frac {4\pi }{3}}}$	${\displaystyle -{\frac {\sqrt {3}}{2}}}$	${\displaystyle -{\frac {1}{2}}}$	${\displaystyle {\sqrt {3}}}$	${\displaystyle {\frac {\sqrt {3}}{3}}}$
270	${\displaystyle {\frac {3\pi }{2}}}$	${\displaystyle -1\,}$	${\displaystyle 0\,}$	${\displaystyle -\,}$	${\displaystyle 0\,}$
300	${\displaystyle {\frac {5\pi }{3}}}$	${\displaystyle -{\frac {\sqrt {3}}{2}}}$	${\displaystyle {\frac {1}{2}}}$	${\displaystyle -{\sqrt {3}}}$	${\displaystyle -{\frac {\sqrt {3}}{3}}}$
315	${\displaystyle {\frac {7\pi }{4}}}$	${\displaystyle -{\frac {\sqrt {2}}{2}}}$	${\displaystyle {\frac {\sqrt {2}}{2}}}$	${\displaystyle -1\,}$	${\displaystyle -1\,}$
330	${\displaystyle {\frac {11\pi }{6}}}$	${\displaystyle -{\frac {1}{2}}}$	${\displaystyle {\frac {\sqrt {3}}{2}}}$	${\displaystyle {\frac {-{\sqrt {3}}}{3}}}$	${\displaystyle -{\sqrt {3}}}$

Související články

• Goniometrie
• Goniometrické funkce
• Kvadrant (geometrie)
• Jednotková kružnice
• Substituce (matematika)
• Rovnice
• Kvadratická rovnice