Given: The cosine function below
[tex]\cos \theta=\frac{12}{13}[/tex]
To Determine: the value of
[tex]\sin 2\theta[/tex]
Given that the angle is acute, we can represent this using a right-angles triangle to determine the dimension of the third side
Using the Pythagoras theorem, we can get the value of the opposite sides
[tex]\begin{gathered} a^2+12^2=13^2,a=\text{opposite} \\ a^2+144=169 \\ a^2=169-144 \\ a^2=25 \\ a=\sqrt[]{25}=5 \end{gathered}[/tex][tex]\sin \theta=\frac{5}{13}[/tex]
Using trigonometry identities,
[tex]\sin 2\theta=2sin\theta\cos \theta[/tex]
Substituting into the identities
[tex]\sin 2\theta=2\times\frac{5}{13}\times\frac{12}{13}=\frac{120}{169}[/tex]
Hence,
[tex]\sin 2\theta=\frac{120}{169}[/tex]
