Determine the values of x,y such that

mwombenizhjb

mwombenizhjb

Answered question

2022-03-28

Determine the values of x,y such that cosxcoszcosy+sinxsiny<0 for all z

Answer & Explanation

Jambrichp2w2

Jambrichp2w2

Beginner2022-03-29Added 12 answers

Write
x=jπ+ξ,y=kπ+η,|ξ|,|η|π2
Then the condition
sinxsiny<tcosxcosy(1t1)
translates into
(1)j+ksinξsinη<(1)j+ktcosξcosη(1t1)
I shall do the case j+k even and assume |ξ|, |η|<π2 for simplicity. Then we want
tanξtanη<t(1t1)
which is tantamount to tanξtanη<1 , so that ξ  and  η necessarily have different sign 0. If ξ<0<η we obtain the condition tan|ξ|tanη<1, or
tan|ξ|>cotη=tan(π2η)
This implies |ξ|>π2η, and leads to a triangular shape in the second quadrant. When η<0<ξ we obtain this shape reflected in the fourth quadrant.
The case j+k odd has to be treated in the same way.

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