NCERT Example 2 — Introduction to Trigonometry

NCERT Example

Example 2

equal sine ratios · similar triangles

If B and Q are acute angles such that sin B = sin Q, prove that B = Q.

Board-exam working:

Consider right triangles ABC and PQR as shown.

\sin B=\frac{AC}{AB},\qquad \sin Q=\frac{PR}{PQ}

\frac{AC}{AB}=\frac{PR}{PQ}

\frac{AC}{PR}=\frac{AB}{PQ}=k

By the Pythagoras theorem:

BC=\sqrt{AB^2-AC^2},\qquad QR=\sqrt{PQ^2-PR^2}

\frac{BC}{QR}=\frac{\sqrt{k^2PQ^2-k^2PR^2}}{\sqrt{PQ^2-PR^2}}=k

\frac{AC}{PR}=\frac{AB}{PQ}=\frac{BC}{QR}

Therefore, △ACB ∼ △PRQ by SSS similarity. Corresponding angles are equal.

\angle B=\angle Q

Hence, ∠B = ∠Q.