If [tex]\cos x = \frac{1}{12}[/tex] and [tex]\sin x > 0[/tex], find [tex]\tan 2x[/tex].

cos x = 12 1 ​ s i n 2 x + co s 2 x = 1 s i n 2 x + ( 12 1 ​ ) 2 = 1 s i n 2 x + 144 1 ​ = 1 s i n 2 x = 1 − 144 1 ​ s i n 2 x = 144 144 ​ − 144 1 ​ s i n 2 x = 144 143 ​ s in x = 144 143 ​ ​ s in x = 12 143 ​ ​
t an 2 x = cos 2 x s in 2 x ​ s in 2 x = 2 s in x cos x ; cos 2 x = co s 2 x − s i n 2 x s in 2 x = 2 ⋅ 12 143 ​ ​ ⋅ 12 1 ​ = 144 2 143 ​ ​ cos 2 x = ( 12 1 ​ ) 2 − ( 12 143 ​ ​ ) 2 = 144 1 ​ − 144 143 ​ = − 144 142 ​ t an 2 x = 144 2 143 ​ ​ : ( − 144 142 ​ ) = − 144 2 143 ​ ​ ⋅ 142 144 ​ = − 71 143 ​ ​

Answered by Anonymous | 2024-06-10

Given cos x = 12 1 ​ and 0"> sin x > 0 , we find that tan 2 x = − 71 143 ​ ​ . This is derived through trigonometric identities and relationships. The calculations involve finding sin x before applying the double angle formulas for sine and cosine.
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Answered by Anonymous | 2024-12-20