If the data given to construct a triangle ABC are a = 5, b = 7, \(\sin A = \frac{3}{4}\), then it is possible to construct

Concept:

Sine (sin θ) is the function revealing the shape of a right triangle. Looking out from a vertex with angle θ, sinθ is the ratio of the opposite side to the hypotenuse in a right-angled triangle.  Also, we know In any right-angled 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,

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 cosine

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

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

⇒ 2c² + 48 - \(7\sqrt{7}\) c = 0

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

⇒ D = 343 - 384 < 0

Hence, the roots are not real. Therefore, No triangle is possible because the side of the triangle cannot be imaginary.