First slide

Introduction to Determinants

Question

In a triangle, $A B C$ if $Δ = |\begin{array}{lll} 1 a b \\ 1 c a \\ 1 b c \end{array}| = 0$ then $\sin^{2} A + \sin^{2} B + \sin^{2} C$ is

Moderate

A

$\frac{3 \sqrt{3}}{2}$
B

$\frac{5}{4}$
C

$\frac{9}{4}$
D

2

Solution

Evaluating along $C_{1}$ , we get
$\begin{array}{ll} Δ = (c^{2} - a b) + (b^{2} - a c) + (a^{2} - b c) = 0 \\ \Rightarrow (a - b)^{2} + (b - c)^{2} + (c - a)^{2} = 0 \\ \Rightarrow a = b = c \Rightarrow A = B = C = π / 3 \\ ∴ \sin^{2} A + \sin^{2} B + \sin^{2} C = \frac{3 \sqrt{3}}{2} \end{array}$

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