Question

what is \\(\sin d\\)?
\\(\sin d = \square\\)

Explanation:

Step1: Identify Similar Triangles

Triangles \(ABC\) and \(DFE\) are right - angled triangles (\(\angle C=\angle F = 90^{\circ}\)). If they are similar, corresponding angles are equal. So \(\angle D=\angle A\) (or we can analyze the trigonometric ratios based on the sides of triangle \(ABC\) since the triangles are similar in structure).

Step2: Recall Sine Definition

In a right - angled triangle, \(\sin\theta=\frac{\text{opposite side}}{\text{hypotenuse}}\). For angle \(A\) in triangle \(ABC\), the opposite side to \(\angle A\) is \(BC = 12\) and the hypotenuse is \(AB=15\). Since \(\angle D\) has the same trigonometric ratio as \(\angle A\) (because the triangles are similar, corresponding angles are equal and trigonometric ratios depend on the angle), we calculate \(\sin D=\sin A\).
\(\sin A=\frac{BC}{AB}=\frac{12}{15}\)

Step3: Simplify the Fraction

Simplify \(\frac{12}{15}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 3. \(\frac{12\div3}{15\div3}=\frac{4}{5}\)

Answer:

\(\frac{4}{5}\)