How to solve 2 sin 2 </msup> &#x2061;<!-- ⁡ --> ( x ) +

Jay Barrett

Jay Barrett

Answered question

2022-05-14

How to solve 2 sin 2 ( x ) + cos 2 ( x ) + cos ( x ) = 0

Answer & Explanation

Agrebbef4l93

Agrebbef4l93

Beginner2022-05-15Added 6 answers

You can rewrite the equation to
2 ( 1 cos 2 x ) + cos 2 x + cos x = 0
Now, introduce a new variable: y = cos x and first solve for y.
Then, for each solution y, every x that solves the equation cos x = y solves the original equation.
britesoulusjhq

britesoulusjhq

Beginner2022-05-16Added 2 answers

You can write the equation
s i n 2 ( x ) + s i n 2 ( x ) + c o s 2 ( x ) + c o s ( x ) = 1 + s i n 2 ( x ) + c o s ( x ) = 0 But clearly 1 1 + s i n 2 ( x ) and this implies 1 + c o s ( x ) 1 + s i n 2 ( x ) + c o s ( x ). The right hand side of this inequality might be zero only if c o s ( x ) = 1 and in this case we have s i n ( x ) = 0 that gives 1 + s i n 2 ( x ) + c o s ( x ) = 0. So the solutions are when c o s ( x ) = 1 i.e. x = ( 2 k + 1 ) π

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