Concept:

Sine (sin θ) is a function that represents the shape of a right triangle. Looking from a vertex with angle θ, sinθ in a right triangle is the ratio of the opposite side to the hypotenuse. Additionally, we know that in any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Therefore, we need to find the third side of the triangle.

Calculation:

a = 5, b = 7, sin A = 3/4

We know that,

sin²A + cos²A = 1

⇒ cos A = \(\sqrt{1 - sin^2 A}\)

⇒ cos A = \(\sqrt{1 - \frac{9}{16}}\)

⇒ cos A = \(\sqrt{\frac{7}{16}}\)

⇒ cos A = \(\frac{\sqrt{7}}{4}\)

Formula: Law of Cosines

\(cos A = \frac{b^2 + c^2 - a^2}{2 bc}\) , where c = length of side c, a = length of side a, b = length of side b, and A = the angle opposite to c

⇒ \(\frac{\sqrt{7}}{4}\) = \(\frac{49 + c^2 - 25}{2 (7)c}\)

⇒ 2c² + 48 - \(7\sqrt{7}\) c = 0

To determine the nature of the roots of the quadratic equation, we need D = \(\sqrt{b^2-4ac}\)

⇒ D = 343 - 384 < 0

Thus, the roots are not real. Therefore, no triangle is possible since the sides of a triangle cannot be imaginary.