In triangle \(ABC\), angle \(ACB=90^\circ\), \(AB=15.4\) cm and angle \(BAC=52^\circ\). Calculate the length of \(BC\), correct to \(3\) significant figures...
Question 1 Report
In triangle \(ABC\), angle \(ACB=90^\circ\), \(AB=15.4\) cm and angle \(BAC=52^\circ\).
Calculate the length of \(BC\), correct to \(3\) significant figures.
Answer Details
The right angle is at \( C \), so \( AB = 15.4 \) cm is the hypotenuse. Relative to the \( 52^\circ \) angle at \( A \), the side \( BC \) is opposite, so the ratio of opposite to hypotenuse is required, which is the sine.
\[ \sin 52^\circ = \frac{BC}{15.4} \] \[ BC = 15.4 \times \sin 52^\circ \] [M1] \[ = 15.4 \times 0.788011\ldots = 12.135\ldots \]To \( 3 \) significant figures, \( BC = 12.1 \) cm. [A1]
Identifying the hypotenuse first is the key step: it is always the side facing the right angle, here \( AB \), never a side of the right angle itself. Since a sine is always less than \( 1 \), \( BC \) must be shorter than the hypotenuse, and \( 12.1 \lt 15.4 \) confirms this.
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