Тригонометрични уравнения

[Гост]

1. Sinx= 3/5
x[pi/2]\epsilon[/[tex]\pi[/tex]] Търси се: cosx , cotgx, tgx = ?

2.Докажете тъждеството за стойността на \alpha , при които и двете му страни са дефинирани:
a) \frac{1+cotga}{1-cotga} = \frac{tga+1}{tga-1} б) \frac{tga}{1-tg^2\alpha} . \frac{cotg^2\alpha -1}{cotga}=1
в) tg^2a - sin^2a = tg^2a . sin^2a г) \frac{sin^2a-cos^2a+cos^4a}{cos^2a-sin^2a+sin^4a} = tg^4a

[Гост]

1. Sinx= 3/5
x[pi/2]\epsilon[/[tex]\pi[/tex]] Търси се: cosx , cotgx, tgx = ?

2.Докажете тъждеството за стойността на \alpha , при които и двете му страни са дефинирани:
a) \frac{1+cotga}{1-cotga} = \frac{tga+1}{tga-1} б) \frac{tga}{1-tg^2\alpha} . \frac{cotg^2\alpha -1}{cotga}=1
в) tg^2a - sin^2a = tg^2a . sin^2a г) \frac{sin^2a-cos^2a+cos^4a}{cos^2a-sin^2a+sin^4a} = tg^4a

[mail_dinko]

1.
[tex]sinx= 3/5[/tex]
[tex]x \in ( \frac {\pi}{2} ; \pi )[/tex]
[tex]sin^2 x + cos^2 x =1 \Rightarrow cos x = \sqrt {1- sin^2 x} = \sqrt {1- \frac {9}{25}} = \sqrt {\frac {25}{25}- \frac {9}{25}} \Rightarrow cos x = \sqrt { (\pm \frac 45)^2}[/tex]
[tex]\fbox {cosx = - \frac 45}[/tex]
[tex]tg x = \frac {sinx}{cosx}= - \frac {3\cancel {.5}}{\cancel{5}.4} \Rightarrow \fbox {tgx = - \frac 34 }[/tex]
[tex]cotg x = \frac {1}{tgx} \Rightarrow \fbox {cotg x = - \frac 43}[/tex]

2. a)
[tex]\frac{1+cotga}{1-cotga} = \frac{tga+1}{tga-1}[/tex]
Умножаваме на кръст
[tex](1+cotg a )(tga-1)=(tga+1)(1-cotga)[/tex]
[tex]tga-1+cotg a . tg a - cotg a = tg a - tg a . cotg a + 1 - cotg a[/tex]
[tex]tga-1+1 - cotg a = tg a - 1 + 1 - cotg a[/tex]
[tex]tga - cotg a = tg a - cotg a[/tex]

2. б)
[tex]\frac{tga}{1-tg^2\alpha} . \frac{cotg^2\alpha -1}{cotga}=1[/tex]
[tex]\frac{tga}{1- \frac {sin ^2 a}{cos^2 a}} . \frac{ \frac {cos^2 a}{sin ^2 a}-1}{cotga}=1[/tex]
[tex]\frac{tga}{ \frac {cos ^2 a}{cos^2 a}- \frac {sin ^2 a}{cos^2 a}} . \frac{ \frac {cos^2 a}{sin ^2 a}- \frac {sin ^2 a}{sin^2 a}}{cotga}=1[/tex]
[tex]\frac {tg a}{cotg a} . \frac {\cancel {cos ^2 a - sin ^2 a}}{sin ^2 a} . \frac {cos^2 a}{\cancel {cos^2 a - sin ^2 a}} =\frac {tg a}{\cancel {cotg a}} . cotg ^{\cancel {2}} a = tg a . cotg a = 1[/tex]

2. в)
[tex]tg^2a - sin^2a = tg^2a . sin^2a[/tex]
[tex]\frac {sin ^2 a}{cos^2 a} - sin^2 a =\frac {sin ^2 a}{cos^2 a} -\frac {sin ^2 a . cos^2 a}{cos^2 a} = \frac {sin^2 a - sin ^2 a . cos^2 a}{cos^2 a } =[/tex]
[tex]= \frac {sin^2 a (1 - cos^2 a)}{cos^2 a } = tg ^2 a . sin ^2 a[/tex]

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