5 задачи за тригонометрични функции!

[Гост]

1.[tex]\frac{cos2\alpha }{ 1-sin2\alpha} = cotg(\frac{\pi }{4 } -\alpha )[/tex]

2. [tex](sin\alpha + sin\beta)^2 + (cos\alpha+cos\beta)^2 = 4cos^2 \frac{ \alpha-\beta}{2}[/tex] (това е 4 косинус квадрат, не 4cos2)

3.[tex]tg\alpha +tg(\pi-\alpha) +cotg(\frac{\pi}{2} + \alpha) = tg(2\pi - \alpha)[/tex]

4.[tex]\frac {sin^4 \alpha + cos^4 \alpha - 1}{sin^2 \alpha. cos^2 \alpha} = -2[/tex]

5.[tex]\frac {sin^8 \alpha + cost^8 \alpha -1}{sin^2 \alpha.cos^2 \alpha} + sin^4 \alpha + cos^4 \alpha = -3[/tex]

Ако успеете да ги решите Ви благодаря предварително :p !

[mail_dinko]

1.
[tex]\frac{cos2\alpha }{ 1-sin2\alpha} = cotg(\frac{\pi }{4 } -\alpha )[/tex]
[tex]\frac{cos2\alpha }{ 1-sin2\alpha} =\frac {cotg 45 ^\circ cotg \alpha +1}{cotg \alpha - cotg 45 ^ \circ}[/tex]
[tex]\frac{cos^2 \alpha - sin^2 \alpha}{sin^2 \alpha + cos^2 \alpha -2 sin \alpha cos \alpha}=\frac { cotg \alpha +1}{cotg \alpha - 1}[/tex]
[tex]\frac {\cancel{(cos \alpha - sin \alpha)}(cos \alpha + sin \alpha)}{(cos \alpha - sin \alpha)^{\cancel {2}}}=\frac {\frac {cos \alpha}{sin \alpha}+1}{\frac {cos \alpha}{sin \alpha}-1}[/tex]
[tex]\frac {cos \alpha + sin \alpha}{cos \alpha - sin \alpha}=\frac {\frac {cos \alpha}{sin \alpha}+\frac {sin \alpha}{sin \alpha}}{\frac {cos \alpha}{sin \alpha}-\frac {sin \alpha}{sin \alpha}}[/tex]
[tex]\frac {cos \alpha + sin \alpha}{cos \alpha - sin \alpha}=\frac {cos \alpha + sin \alpha}{\cancel {sin \alpha}}.\frac {\cancel {sin \alpha}}{cos \alpha - sin \alpha[/tex]
[tex]\frac {cos \alpha + sin \alpha}{cos \alpha - sin \alpha}=\frac {cos \alpha + sin \alpha}{cos \alpha - sin \alpha}[/tex]

2.
[tex](sin\alpha + sin\beta)^2 + (cos\alpha+cos\beta)^2 = 4cos^2 \frac{ \alpha-\beta}{2}[/tex]
[tex](sin\alpha + sin\beta)^2 + (cos\alpha+cos\beta)^2 =[/tex]
[tex]=(2sin {\frac {\alpha + \beta }{2}}cos {\frac {\alpha - \beta}{2}})^2+(2cos {\frac {\alpha + \beta }{2}}cos {\frac {\alpha - \beta}{2}})^2=[/tex]
[tex]=4sin^2 {\frac {\alpha + \beta }{2}}cos^2 {\frac {\alpha - \beta}{2}}+4cos^2 {\frac {\alpha + \beta }{2}}cos^2 {\frac {\alpha - \beta}{2}}=[/tex]
[tex]=4cos^2 {\frac {\alpha - \beta}{2}}(sin^2 {\frac {\alpha + \beta }{2}}+cos^2 {\frac {\alpha + \beta }{2}})=[/tex]
[tex]=4cos^2 {\frac {\alpha - \beta}{2}}[/tex]

След малко ще напиша и другите

[mail_dinko]

3.
[tex]tg\alpha +tg(\pi-\alpha) +cotg(\frac{\pi}{2} + \alpha) = tg(2\pi - \alpha)[/tex]
[tex]\cancel{tg\alpha} \cancel{-tg\alpha}-tg\alpha= -tg\alpha[/tex]
[tex]-tg\alpha= -tg\alpha[/tex]

4.
[tex]\frac {sin^4 \alpha + cos^4 \alpha - 1}{sin^2 \alpha. cos^2 \alpha} = -2[/tex]
[tex]\frac {(sin^2 \alpha + cos^2 \alpha )^2 - 2 sin^2 \alpha cos ^2 \alpha - 1}{sin^2 \alpha. cos^2 \alpha} =[/tex]
[tex]=\frac {\cancel{1}-2sin^2 \alpha cos^2 \alpha \cancel{-1}}{sin^2 \alpha cos^2 \alpha}=[/tex]
[tex]=\frac {-2 \cancel {sin^2 \alpha cos^2 \alpha}}{\cancel{sin^2 \alpha cos^2 \alpha}}=[/tex]
[tex]=-2[/tex]

[mail_dinko]

5.
[tex]\frac {sin^8 \alpha + cost^8 \alpha -1}{sin^2 \alpha.cos^2 \alpha} + sin^4 \alpha + cos^4 \alpha = -3[/tex]
[tex]\frac {sin^8 \alpha + cost^8 \alpha -1}{sin^2 \alpha.cos^2 \alpha} + sin^4 \alpha + cos^4 \alpha =[/tex]
[tex]=\frac {(sin^2 \alpha + cos^2 \alpha)(sin^4 \alpha - sin^2 \alpha cos ^2 \alpha + cos^4 \alpha) -1}{sin^2 \alpha.cos^2 \alpha} + sin^4 \alpha + cos^4 \alpha =[/tex]
[tex]=\frac {sin^4 \alpha - sin^2 \alpha cos ^2 \alpha + cos^4 \alpha -1}{sin^2 \alpha.cos^2 \alpha} + sin^4 \alpha + cos^4 \alpha =[/tex]
[tex]=\frac {sin^4 \alpha+ cos^4 \alpha - sin^2 \alpha cos ^2 \alpha -1}{sin^2 \alpha.cos^2 \alpha} + sin^4 \alpha + cos^4 \alpha =[/tex]
[tex]=\frac {(sin^2 \alpha+ cos^2 \alpha)^2 - 3sin^2 \alpha cos ^2 \alpha -1}{sin^2 \alpha.cos^2 \alpha} + sin^4 \alpha + cos^4 \alpha =[/tex]
[tex]=\frac {\cancel{1} - 3sin^2 \alpha cos ^2 \alpha \cancel{-1}}{sin^2 \alpha.cos^2 \alpha} + sin^4 \alpha + cos^4 \alpha =[/tex]
[tex]=\frac { - 3 \cancel{ sin^2 \alpha cos ^2 \alpha }}{\cancel{sin^2 \alpha.cos^2 \alpha}} + sin^4 \alpha + cos^4 \alpha =[/tex]
[tex]=-3+ sin^4 \alpha + cos^4 \alpha=[/tex]
[tex]=-3+(sin^2 \alpha + cos^2 \alpha )^2 -2sin ^2 \alpha cos^2 \alpha=[/tex]
[tex]=-3+1-2sin^2 \alpha cos^2 \alpha=[/tex]
[tex]=-2-2sin^2 \alpha cos^2 \alpha[/tex]

нещо не ми се получава

[ganka simeonova]

5.
[tex]\frac {sin^8 \alpha + cost^8 \alpha -1}{sin^2 \alpha.cos^2 \alpha} + sin^4 \alpha + cos^4 \alpha = -3[/tex]
[tex]\frac {sin^8 \alpha + cost^8 \alpha -1}{sin^2 \alpha.cos^2 \alpha} + sin^4 \alpha + cos^4 \alpha =[/tex]
[tex]=\frac {(sin^2 \alpha + cos^2 \alpha)(sin^4 \alpha - sin^2 \alpha cos ^2 \alpha + cos^4 \alpha) -1}{sin^2 \alpha.cos^2 \alpha} + sin^4 \alpha + cos^4 \alpha =[/tex]


нещо не ми се получава

Няма как да се получи, понеже третият ред е грешен.

[mail_dinko]

5.
[tex]\frac {sin^8 \alpha + cos^8 \alpha -1}{sin^2 \alpha.cos^2 \alpha} + sin^4 \alpha + cos^4 \alpha = -3[/tex]
[tex]\frac {sin^8 \alpha + cos^8 \alpha -1}{sin^2 \alpha.cos^2 \alpha} + sin^4 \alpha + cos^4 \alpha =[/tex]
[tex]=\frac {(sin^4 \alpha)^2 + (cos^4 \alpha)^2-1}{sin^2 \alpha.cos^2 \alpha} + sin^4 \alpha + cos^4 \alpha =[/tex]
[tex]=\frac {(sin^4 \alpha + cos^4 \alpha)^2-2sin^4 \alpha cos^4 \alpha -1}{sin^2 \alpha.cos^2 \alpha} + sin^4 \alpha + cos^4 \alpha =[/tex]
[tex]=\frac {[(sin^2 \alpha + cos^2 \alpha)^2-2sin^2 \alpha cos ^2 \alpha]^2-2sin^4 \alpha cos^4 \alpha -1}{sin^2 \alpha.cos^2 \alpha} + sin^4 \alpha + cos^4 \alpha =[/tex]
[tex]=\frac {[1-2sin^2 \alpha cos ^2 \alpha]^2-2sin^4 \alpha cos^4 \alpha -1}{sin^2 \alpha.cos^2 \alpha} + sin^4 \alpha + cos^4 \alpha =[/tex]
[tex]=\frac {\cancel{1}-4sin^2 \alpha cos ^2 \alpha+4sin^4 \alpha cos^4 \alpha-2sin^4 \alpha cos^4 \alpha \cancel{-1}}{sin^2 \alpha.cos^2 \alpha} + sin^4 \alpha + cos^4 \alpha =[/tex]
[tex]=\frac {2sin^4 \alpha cos^4 \alpha-4sin^2 \alpha cos ^2 \alpha}{sin^2 \alpha.cos^2 \alpha} + sin^4 \alpha + cos^4 \alpha =[/tex]
[tex]=\frac {2\cancel{sin^2 \alpha cos^2 \alpha}(sin^2 \alpha cos^2 \alpha-2)}{\cancel{sin^2 \alpha.cos^2 \alpha}} + sin^4 \alpha + cos^4 \alpha =[/tex]
[tex]=\cancel{2sin^2 \alpha cos ^2 \alpha}-4 +1\cancel{- 2sin^2 \alpha cos^2 \alpha}=[/tex]
[tex]=-4+1=[/tex]
[tex]=-3[/tex]

[ganka simeonova]

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