How can we bound sin &#x2061;<!-- ⁡ --> ( &#x03B8;<!-- θ -->

ownerweneuf

ownerweneuf

Answered question

2022-05-21

How can we bound sin ( θ ) cos ( θ ) | cos ( θ ) | + | sin ( θ ) |

Answer & Explanation

Annabella Velez

Annabella Velez

Beginner2022-05-22Added 7 answers

We know
( | c o s θ | + | s i n θ | ) 2 = 1 + 2 | c o s θ s i n θ | = 1 + | s i n ( 2 θ ) |
So
sin ( θ ) cos ( θ ) | cos ( θ ) | + | sin ( θ ) | = s i n ( 2 θ ) 2 1 + | s i n ( 2 θ ) |
So the bound is given by the maximum of f ( x ) = x 2 1 + x for 0 x 1. F(x) is monotonne increasing, so the maximum is f ( 1 ) = 1 2 2 = 2 4
Therefore,
| sin ( θ ) cos ( θ ) | cos ( θ ) | + | sin ( θ ) | | 2 4
The equality holds at θ = π 4

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