тригонометрични функции

[openfire]

Здравейте!Нуждая се от помощта Ви за тeзи задачи.

[ammornil]

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

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

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

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

[ammornil]

2) Преди да покажа едно възможно опростяване искам да покажа нещо което ще използвам по-долу:
[tex]1-cos^{^{4}}\alpha=(1-cos^{^{2}}\alpha).(1+cos^{^{2}}\alpha)=sin{^{2}}\alpha.(1+1-sin{^{2}}\alpha)=sin{^{2}}\alpha.(2-sin{^{2}}\alpha)=[/tex]
[tex]=2.sin{^{2}}\alpha-sin{^{4}}\alpha[/tex]

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

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

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

[ammornil]

4) [tex](sin\alpha+sin\beta).(sin\alpha-sin\beta)+(cos\alpha+cos\beta).(cos\alpha-cos\beta)=[/tex]

=[tex]sin^{^{2}}\alpha-sin^{^{2}}\beta+cos^{^{2}}\alpha-cos^{^{2}}\beta=(sin^{^{2}}\alpha+cos^{^{2}}\alpha)-(sin^{^{2}}\beta+cos^{^{2}}\beta)=1-1=0[/tex]

[ammornil]

5)[tex]cos\alpha.(1+\frac{1}{cos\alpha}+tg\alpha).(1-\frac{1}{cos\alpha}+tg\alpha)=[/tex]

[tex]=cos\alpha.\left[(1+tg\alpha)+\frac{1}{cos\alpha})\right].\left[(1+tg\alpha)-\frac{1}{cos\alpha})\right]= cos\alpha.\left[(1+tg\alpha)^{^{2}}-(\frac{1}{cos\alpha})^{^{2}}\right]=[/tex]

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

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

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

[ammornil]

3) [tex](tg\alpha+cotg\beta).(tg\alpha-cotg\beta)-(\frac{1}{cos\alpha}+\frac{1}{cos\beta}).(\frac{1}{cos\alpha}-\frac{1}{cos\beta})=[/tex]

[tex]=(tg^{^{2}}\alpha-cotg^{^{2}}\beta)-\left[(\frac{1}{cos\alpha})^{^{2}}-(\frac{1}{cos\beta})^{^{2}}\right]= \frac{sin^{^{2}}\alpha}{cos^{^{2}}\alpha}-\frac{cos^{^{2}}\beta}{sin^{^{2}}\beta}-\left(\frac{1}{cos^{^{2}}\alpha}-\frac{1}{cos^{^{2}}\beta}\right)=[/tex]

[tex]=\frac{1-cos^{^{2}}\alpha}{cos^{^{2}}\alpha}-\frac{cos^{^{2}}\beta}{1-cos^{^{2}}\beta}-\left(\frac{cos^{^{2}}\beta-cos^{^{2}}\alpha}{cos^{^{2}}\alpha.cos^{^{2}}\beta}\right)=\frac{(1-cos^{^{2}}\alpha).(1-cos^{^{2}}\beta)-cos^{^{2}}\alpha.cos^{^{2}}\beta}{cos^{^{2}}\alpha.(1-cos^{^{2}}\beta)}-\left(\frac{cos^{^{2}}\beta-cos^{^{2}}\alpha}{cos^{^{2}}\alpha.cos^{^{2}}\beta}\right)=[/tex]

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

[tex]=\frac{cos^{^{2}}\beta.(1-cos^{^{2}}\beta-cos^{^{2}}\alpha)-(1-cos^{^{2}}\beta).(cos^{^{2}}\beta-cos^{^{2}}\alpha)}{cos^{^{2}}\alpha.cos^{^{2}}\beta.(1-cos^{^{2}}\beta)}=[/tex]

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

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

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

[openfire]

Благодаря!