Question

assignment 7.3: double - angle, half - angle, and reduction score: 0/60 answered: 0/3 question 1 if sin x = 5/9, x in quadrant i, then find (without finding x): sin(2x)= cos(2x)= tan(2x)= question help: video 1 video 2 post to forum

Explanation:

Step1: Find cos(x)

Using $\sin^{2}x+\cos^{2}x = 1$, so $\cos x=\sqrt{1 - \sin^{2}x}$. Since $x$ is in quadrant I, $\cos x>0$. Given $\sin x=\frac{5}{9}$, then $\cos x=\sqrt{1 - (\frac{5}{9})^{2}}=\sqrt{1-\frac{25}{81}}=\sqrt{\frac{81 - 25}{81}}=\sqrt{\frac{56}{81}}=\frac{2\sqrt{14}}{9}$.

Step2: Find sin(2x)

Use the double - angle formula $\sin(2x)=2\sin x\cos x$. Substitute $\sin x=\frac{5}{9}$ and $\cos x=\frac{2\sqrt{14}}{9}$, we get $\sin(2x)=2\times\frac{5}{9}\times\frac{2\sqrt{14}}{9}=\frac{20\sqrt{14}}{81}$.

Step3: Find cos(2x)

Use the double - angle formula $\cos(2x)=1 - 2\sin^{2}x$. Substitute $\sin x=\frac{5}{9}$, we have $\cos(2x)=1-2\times(\frac{5}{9})^{2}=1 - 2\times\frac{25}{81}=1-\frac{50}{81}=\frac{31}{81}$.

Step4: Find tan(2x)

Use the formula $\tan(2x)=\frac{\sin(2x)}{\cos(2x)}$. Substitute $\sin(2x)=\frac{20\sqrt{14}}{81}$ and $\cos(2x)=\frac{31}{81}$, we obtain $\tan(2x)=\frac{\frac{20\sqrt{14}}{81}}{\frac{31}{81}}=\frac{20\sqrt{14}}{31}$.

Answer:

$\sin(2x)=\frac{20\sqrt{14}}{81}$
$\cos(2x)=\frac{31}{81}$
$\tan(2x)=\frac{20\sqrt{14}}{31}$