Respuesta :
Explanation
We are given the following:
[tex]\sin\theta=-\frac{7}{25}\text{ }and\text{ }\pi<\theta<\frac{3\pi}{2}[/tex]We are required to determine the exact value of sin 2Θ.
We know that the trigonometric identity for sin 2Θ is thus:
[tex]\sin2\theta=2\sin\theta\cos\theta[/tex]Since Θ is between 180° and 270° as given above, we know that this angle falls in the third quadrant, and sine and cosine are negative in this quadrant.
Therefore, we have:
[tex]\begin{gathered} \sin\theta=\frac{7}{25}\to\frac{opposite}{hypotenuse} \\ \\ \text{ Using the Pythagorean theorem,} \\ hypotenuse^2=opposite^2+adjacent^2 \\ 25^2=7^2+adj^2 \\ adj^2=25^2-7^2 \\ adj=\sqrt{25^2-7^2} \\ adj=\sqrt{625-49}=\sqrt{576} \\ adj=24 \\ \\ \text{ Hence, we have:} \\ \cos\theta=\frac{adjacent}{hypotenuse} \\ \cos\theta=\frac{24}{25} \\ \\ \text{ In the third quadrant, } \\ \cos\theta=-\frac{24}{25} \end{gathered}[/tex]Now, we can determine the value of sin 2Θ as:
[tex]\begin{gathered} \sin2\theta=2\sin\theta\cos\theta \\ \sin2\theta=2\cdot(-\frac{7}{25})\cdot(-\frac{24}{25}) \\ \sin2\theta=\frac{2\times7\times24}{25\times25} \\ \sin2\theta=\frac{336}{625} \end{gathered}[/tex]Hence, the answer is:
[tex]\sin2\theta=\frac{336}{625}[/tex]