已知tan a / tan a-1= -1,求下列各式的值

题目

已知tan a / tan a-1= -1,求下列各式的值
1) sin a-3cos a / sin a + cos a
2) sin a^2+ sin a cos a + 2

答案

tan a / (tan a-1)= -1
tan a =1-tan a
2tan a =1
tan a =1/2
(sin a-3cos a) / (sin a + cos a)分子分母同时除以cos a
=(sin a/cos a-3cos a/cos a) / (sin a/cos a + cos a/cos a)
=(tan a -3)/(tan a +1)
=(1/2-3)/(1/2+1)
=(-5/2)/(3/2)
=-5/3
sin a^2+ sin a cos a + 2
=(sin a^2+ sin a cos a + 2)/1
=[sin a^2+ sin a cos a + 2(sin a^2+cos a^2)]/(sin a^2+cos a^2)
=(3sin a^2+ sin a cos a + 2cos a^2)/(sin a^2+cos a^2)分子分母同时除以cos a^2
=(3sin a^2/cos a^2+ sin a cos a/cos a^2 + 2cos a^2/cos a^2)/(sin a^2/cos a^2+cos a^2/cos a^2)
=(3tan a^2+ tan a + 2)/(tan a^2+1)
=[3*(1/2)^2+1/2+2]/[(1/2)^2+1]
=(3/4+1/4+2)/(1/4+1)
=3/(5/4)
=3*4/5
=12/5