Question

若sina+cosb=根号2/2,求cosa+cosb的取值范围

Answer 1

sina+cosb=√2/2

cosb=√2/2-sina

令y=cosa+cosb

y=cosa+√2/2-sina

=√2(√2/2cosa-√2/2sina)+√2/2

=√2cos(a+π/4)+√2/2

因为cos(a+π/4)∈[-1,1]

所以y最大值=√2+√2/2=3√2/2

y最小值=-√2+√2/2=-√2/2

y∈[3√2/2,-√2/2]