Use double angle formulas to find the exact values of \(\sin(2x)\), \(\cos(2x)\), and \(\tan(2x)\).

Given: \(\sin(x) = \frac{12}{13}\), \(0 < x < \frac{\pi}{2}\)

sin2x = 2sinxcosx; cos2x = (cosx)^2 - (sinx)^2; tan2x = (sin2x)/(cos2x);
cosx = 5/13 from formula (sinx)^2 + (cosx)^2 = 1;
=> sin2x = 120/169; .................................

Answered by crisforp | 2024-06-10

Using double angle formulas, we find sin ( 2 x ) = 169 120 ​ , cos ( 2 x ) = − 169 119 ​ , and tan ( 2 x ) = − 119 120 ​ . We calculated these by first finding cos ( x ) from sin ( x ) and then applying the formulas. All calculations followed proper trigonometric identities.
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Answered by crisforp | 2024-12-20