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Mathematics · Introduction to Trigonometry · NCERT Examples

Example 2

A complete, independently verified solution for NCERT Example 2.

Mathematics · Chapter 8 · NCERT Examples

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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.

Two right triangles ABC and PQR with equal acute angles B and Q marked for comparison.
Figure 8-9 from the supplied NCERT chapter.

Solution

Board-exam working:

Consider right triangles ABC and PQR as shown.

sin⁡B=ACAB,sin⁡Q=PRPQ\sin B=\frac{AC}{AB},\qquad \sin Q=\frac{PR}{PQ}
ACAB=PRPQ\frac{AC}{AB}=\frac{PR}{PQ}
ACPR=ABPQ=k\frac{AC}{PR}=\frac{AB}{PQ}=k

By the Pythagoras theorem:

BC=AB2−AC2,QR=PQ2−PR2BC=\sqrt{AB^2-AC^2},\qquad QR=\sqrt{PQ^2-PR^2}
BCQR=k2PQ2−k2PR2PQ2−PR2=k\frac{BC}{QR}=\frac{\sqrt{k^2PQ^2-k^2PR^2}}{\sqrt{PQ^2-PR^2}}=k
ACPR=ABPQ=BCQR\frac{AC}{PR}=\frac{AB}{PQ}=\frac{BC}{QR}

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

∠B=∠Q\angle B=\angle Q

Final answer

Hence, ∠B = ∠Q.