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Ryan Reynolds

Ryan Reynolds

Answered question

2022-05-22

Let n Z be an integer such that | cos n | < 1 4 . Prove that | cos ( n + 1 ) | > 1 4

Answer & Explanation

Travis Fernandez

Travis Fernandez

Beginner2022-05-23Added 10 answers

By the cosine addition trig identity:
(1) cos ( n + 1 ) = cos n cos 1 sin n sin 1
where
| cos n cos 1 | = cos 1 | cos n | < ( cos 1 ) 1 4 < 0.1351
and since
cos 2 n + sin 2 n = 1 | sin n | > 1 1 16 = 15 4
we also have
| sin n sin 1 | = sin 1 | sin n | > ( sin 1 ) 15 4 > 0.8147
So the first term in equation (1) has magnitude less than 0.1351, while the second term has magnitude greater than 0.8147, hence:
| cos ( n + 1 ) | > 0.8147 0.1351 = 0.6796

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