Answer:
[tex]\sin \text{ 2}\theta\text{ = -}\frac{240}{289}[/tex]
Explanation:
Mathematically, we know that:
[tex]\text{ sin2}\theta\text{ = 2sin}\theta\cos \theta[/tex]
We already have the value for sine (opposite divided by hypotenuse), but we need the cosine value (adjacent divided by hypotenuse)
In light of this, we need the adjacent value of the given triangle
To get this, we use Pythagoras' theorem. It states that the square of the length of the hypotenuse equals the sum of the lengths of the squares of the opposite and the adjacent sides
Let us get the adjacent side as follows:
[tex]\begin{gathered} \text{ 17}^2=15^2+a^2 \\ a^2=17^2-15^2 \\ a^2\text{ = 64} \\ a\text{ = }\sqrt[]{64}\text{ = 8} \end{gathered}[/tex]
The cosine value is thus:
[tex]\text{ cos}\theta\text{ = -}\frac{8}{17}[/tex]
This value is negative because the cosine of an angle is negative on the second quadrant
Substituting the values into the initial equation, we have it that:
[tex]\sin 2\theta\text{ = 2sin}\theta\cos \theta\text{ = 2}\times\frac{15}{17}\times\frac{-8}{17}\text{ = -}\frac{240}{289}[/tex]
