Question

assignment 9.3: double - angle, half - angle formulas
score: 4/13 answered: 2/10
question 3
if \\( \cos x = \frac { 3 } { 5 } \\), \\( x \\) in quadrant i, then find (without finding \\( x \\))
\\( \sin ( 2 x ) = \\)
\\( \cos ( 2 x ) = \\)
question help: video 1 video 2 message instructor
submit question

Explanation:

Step1: Find $\sin x$

Use the identity $\sin^{2}x+\cos^{2}x = 1$.
Given $\cos x=\frac{3}{5}$, then $\sin^{2}x=1-\cos^{2}x=1 - (\frac{3}{5})^{2}=1-\frac{9}{25}=\frac{16}{25}$.
Since $x$ is in quadrant I, $\sin x>0$, so $\sin x=\frac{4}{5}$.

Step2: Calculate $\sin(2x)$

Use the double - angle formula $\sin(2x)=2\sin x\cos x$.
Substitute $\sin x = \frac{4}{5}$ and $\cos x=\frac{3}{5}$ into the formula:
$\sin(2x)=2\times\frac{4}{5}\times\frac{3}{5}=\frac{24}{25}$.

Step3: Calculate $\cos(2x)$

Use the double - angle formula $\cos(2x)=2\cos^{2}x - 1$.
Substitute $\cos x=\frac{3}{5}$ into the formula:
$\cos(2x)=2\times(\frac{3}{5})^{2}-1=2\times\frac{9}{25}-1=\frac{18}{25}-1=-\frac{7}{25}$.

Answer:

$\sin(2x)=\frac{24}{25}$
$\cos(2x)=-\frac{7}{25}$