Solve 2 cos &#x2061;<!-- ⁡ --> 2 x &#x2212;<!-- − --> 5 cos &#x2061;<!--

hawatajwizp

hawatajwizp

Answered question

2022-06-08

Solve 2 cos 2 x 5 cos x 4 = 0 for 0 < x < 2 π

Answer & Explanation

lodosr

lodosr

Beginner2022-06-09Added 24 answers

We have cos 2 x = 2 cos 2 x 1 for all x R , so your equation is equivalent to
2 ( 2 cos 2 x 1 ) 5 cos x 4 = 0
4 cos 2 x 5 cos x 6 = 0 ,
which is a quadratic equation in terms of cosx.
You'll get cos x { 3 4 , 2 } . But by the definition of cosx we have cos x [ 1 , 1 ], so cos x = 3 4 is the only possibility, giving you the only solution x = ± arccos ( 3 4 ) + 2 π k, where k Z
Since you need 0 < x < 2 π, your answer is either x = arccos ( 3 4 ) or x = arccos ( 3 4 ) + 2 π

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