\(\displaystyle{ sin ^{2} x=cosx-1}\)
\(\displaystyle{ tgx+ctgx=2}\)
\(\displaystyle{ 2 sin ^{2} x - sin x - 1=0}\)
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rozwiąż równania
a)
\(\displaystyle{ sin ^{2} x=cosx-1}\)
z jedynki trygonometrycznej możemy zamienić sinusa:
\(\displaystyle{ sin ^{2} x= 1- cos^{2} x}\)
\(\displaystyle{ 1- cos^{2} x=cosx-1}\)
\(\displaystyle{ cos^{2} x+cosx-2 =0}\)
\(\displaystyle{ t=cos x}\) gdzie \(\displaystyle{ Z:t }\)
\(\displaystyle{ t ^{2} + t-2=0}\)
\(\displaystyle{ \Delta=9}\)
\(\displaystyle{ \sqrt{\Delta} = 3}\)
\(\displaystyle{ t _{1} = \frac{-1-3}{2} =-2}\) lub \(\displaystyle{ t _{2} = \frac{-1+3}{2} =1}\)
\(\displaystyle{ t_{1} Z}\)
\(\displaystyle{ cos x = 1}\)
\(\displaystyle{ x=2k \pi}\)
[ Dodano: 15 Maj 2008, 16:06 ]
b)
\(\displaystyle{ tgx+ctgx=2}\)
\(\displaystyle{ \frac{1}{ctg x} +ctg x =2}\)
\(\displaystyle{ \frac{1+ctg ^{2}x }{ctg x} =2}\)
\(\displaystyle{ 1+ctg ^{2}x=2 ctg x}\)
\(\displaystyle{ ctg ^{2}x-2 ctg x+1=0}\)
\(\displaystyle{ ctg x=t}\)
\(\displaystyle{ t ^{2} -2t+1=0}\)
\(\displaystyle{ \Delta=0}\)
\(\displaystyle{ t _{0} = \frac{2}{2} =1}\)
\(\displaystyle{ ctg x=1}\)
\(\displaystyle{ x= \frac{\pi}{4} + k \pi}\)
[ Dodano: 15 Maj 2008, 16:13 ]
c)
\(\displaystyle{ 2 sin ^{2} x - sin x - 1=0}\)
\(\displaystyle{ sin x=t}\) gdzie \(\displaystyle{ Z:t }\)
\(\displaystyle{ 2t ^{2} -t-1=0}\)
\(\displaystyle{ \Delta=9}\)
\(\displaystyle{ \sqrt{\Delta} =3}\)
\(\displaystyle{ t _{1} = \frac{1-3}{4} =- \frac{1}{2}}\) lub \(\displaystyle{ t _{2} = \frac{1+3}{4} =1}\)
\(\displaystyle{ sin x= - \frac{1}{2}}\)
\(\displaystyle{ x =- \frac{\pi}{6} + 2k \pi}\) \(\displaystyle{ \vee}\) \(\displaystyle{ x= \frac{7 \pi}{6} + 2k \pi}\)
lub
\(\displaystyle{ sin x=1}\)
\(\displaystyle{ x= \frac{\pi}{2} +2k \pi}\)
\(\displaystyle{ sin ^{2} x=cosx-1}\)
z jedynki trygonometrycznej możemy zamienić sinusa:
\(\displaystyle{ sin ^{2} x= 1- cos^{2} x}\)
\(\displaystyle{ 1- cos^{2} x=cosx-1}\)
\(\displaystyle{ cos^{2} x+cosx-2 =0}\)
\(\displaystyle{ t=cos x}\) gdzie \(\displaystyle{ Z:t }\)
\(\displaystyle{ t ^{2} + t-2=0}\)
\(\displaystyle{ \Delta=9}\)
\(\displaystyle{ \sqrt{\Delta} = 3}\)
\(\displaystyle{ t _{1} = \frac{-1-3}{2} =-2}\) lub \(\displaystyle{ t _{2} = \frac{-1+3}{2} =1}\)
\(\displaystyle{ t_{1} Z}\)
\(\displaystyle{ cos x = 1}\)
\(\displaystyle{ x=2k \pi}\)
[ Dodano: 15 Maj 2008, 16:06 ]
b)
\(\displaystyle{ tgx+ctgx=2}\)
\(\displaystyle{ \frac{1}{ctg x} +ctg x =2}\)
\(\displaystyle{ \frac{1+ctg ^{2}x }{ctg x} =2}\)
\(\displaystyle{ 1+ctg ^{2}x=2 ctg x}\)
\(\displaystyle{ ctg ^{2}x-2 ctg x+1=0}\)
\(\displaystyle{ ctg x=t}\)
\(\displaystyle{ t ^{2} -2t+1=0}\)
\(\displaystyle{ \Delta=0}\)
\(\displaystyle{ t _{0} = \frac{2}{2} =1}\)
\(\displaystyle{ ctg x=1}\)
\(\displaystyle{ x= \frac{\pi}{4} + k \pi}\)
[ Dodano: 15 Maj 2008, 16:13 ]
c)
\(\displaystyle{ 2 sin ^{2} x - sin x - 1=0}\)
\(\displaystyle{ sin x=t}\) gdzie \(\displaystyle{ Z:t }\)
\(\displaystyle{ 2t ^{2} -t-1=0}\)
\(\displaystyle{ \Delta=9}\)
\(\displaystyle{ \sqrt{\Delta} =3}\)
\(\displaystyle{ t _{1} = \frac{1-3}{4} =- \frac{1}{2}}\) lub \(\displaystyle{ t _{2} = \frac{1+3}{4} =1}\)
\(\displaystyle{ sin x= - \frac{1}{2}}\)
\(\displaystyle{ x =- \frac{\pi}{6} + 2k \pi}\) \(\displaystyle{ \vee}\) \(\displaystyle{ x= \frac{7 \pi}{6} + 2k \pi}\)
lub
\(\displaystyle{ sin x=1}\)
\(\displaystyle{ x= \frac{\pi}{2} +2k \pi}\)