Primeira página
/
Matemática
/
Find sin2x,cos2x and tan2x if sinx=(5)/(13) and x terminates in quadrant I. sin2x= cos2x= tan2x=

Pergunta

Find sin2x,cos2x and tan2x if sinx=(5)/(13) and x terminates in quadrant I.
sin2x=
cos2x=
tan2x=

Find sin2x,cos2x and tan2x if sinx=(5)/(13) and x terminates in quadrant I. sin2x= cos2x= tan2x=

Solução

expert verifiedVerification of experts
4.6119 Voting
avatar
AndréProfissional · Tutor por 6 anos

Responder

Given that $\sin x = \frac{5}{13}$ and $x$ terminates in quadrant I, we can find $\cos x$ using the Pythagorean identity $\sin^2 x + \cos^2 x = 1$. Since $x$ is in quadrant I, $\cos x$ is positive.<br /><br />\[<br />\cos x = \sqrt{1 - \sin^2 x} = \sqrt{1 - \left(\frac{5}{13}\right)^2} = \sqrt{1 - \frac{25}{169}} = \sqrt{\frac{144}{169}} = \frac{12}{13}<br />\]<br /><br />Now, we can find $\sin 2x$, $\cos 2x$, and $\tan 2x$ using the double-angle formulas.<br /><br />\[<br />\sin 2x = 2 \sin x \cos x = 2 \cdot \frac{5}{13} \cdot \frac{12}{13} = \frac{120}{169}<br />\]<br /><br />\[<br />\cos 2x = \cos^2 x - \sin^2 x = \left(\frac{12}{13}\right)^2 - \left(\frac{5}{13}\right)^2 = \frac{144}{169} - \frac{25}{169} = \frac{119}{169}<br />\]<br /><br />\[<br />\tan 2x = \frac{\sin 2x}{\cos 2x} = \frac{\frac{120}{169}}{\frac{119}{169}} = \frac{120}{119}<br />\]<br /><br />Therefore, the values are:<br /><br />\[<br />\sin 2x = \frac{120}{169}<br />\]<br /><br />\[<br />\cos 2x = \frac{119}{169}<br />\]<br /><br />\[<br />\tan 2x = \frac{120}{119}<br />\]
Clique para avaliar: