Answer:
[tex]sin(2\theta)=\frac{\sqrt{35} }{18}[/tex]
Step-by-step explanation:
Recall the formula for the sine of the double angle:
[tex]sin(2\theta)=2*sin(\theta)*cos(\theta)[/tex]
we know that [tex]cos(\theta)=\frac{3}{18}[/tex], and that [tex]\theta[/tex] is in the interval between 0 and 90 degrees, where both the functions sine and cosine are non-negative numbers. Based on such, we can find using the Pythagorean trigonometric property that relates sine and cosine of the same angle, what [tex]sin(\theta)[/tex] is:
[tex]cos^2(\theta)+sin^2(\theta)=1\\sin^2(\theta)=1-cos^2(\theta)\\sin(\theta)=\sqrt{1-cos^2(\theta)} \\sin(\theta)=\sqrt{1-(\frac{3}{18} )^2}\\sin(\theta)=\sqrt{1-\frac{9}{324} }\\sin(\theta)=\sqrt{\frac{324-9}{324} }\\sin(\theta)=\sqrt{\frac{315}{324} }\\\\sin(\theta)=\frac{3}{18}\sqrt{35 }[/tex]
With this information, we can now complete the value of the sine of the double angle requested:
[tex]sin(2\theta)=2*sin(\theta)*cos(\theta)\\sin(2\theta)=2*\frac{3}{18} \,\sqrt{35} \,\frac{3}{18}\\sin(2\theta)=\frac{2*3*3}{18*18}\,\sqrt{35} \\sin(2\theta)=\frac{\sqrt{35} }{18}[/tex]
