The serving bay in the school canteen has a triangular counter \(XYZ\). Angle \(X\) is 96°, the side \(YZ\) measures 7.4 m and the side \(XZ\) measures 5.9 ...
Question 1 Report
The serving bay in the school canteen has a triangular counter \(XYZ\). Angle \(X\) is 96°, the side \(YZ\) measures 7.4 m and the side \(XZ\) measures 5.9 m.
- Calculate the size of angle \(Y\). Give your answer to 1 decimal place. (3)
Answer Details
The known side \(YZ = 7.4\) m lies opposite the known angle \(X = 96\)°, and the required angle \(Y\) lies opposite the known side \(XZ = 5.9\) m. A matched pair of side and opposite angle on both sides of the equation is the signal for the sine rule.
- \(\frac{\sin Y}{5.9} = \frac{\sin 96^\circ}{7.4}\), writing the rule with the sines on top since an angle is being found. [1]
- \(\sin Y = \frac{5.9 \times \sin 96^\circ}{7.4} = \frac{5.9 \times 0.99452\ldots}{7.4} = 0.792929\ldots\). [1]
- \(Y = \sin^{-1}(0.792929\ldots) = 52.4601\ldots\), so \(Y = 52.5\)° to 1 decimal place. [1]
Pair each side with the angle facing it: \(XZ\) joins \(X\) and \(Z\), so the angle opposite it is at \(Y\).
The inverse sine also offers \(180 - 52.5 = 127.5\)°, but that is impossible here because angle \(X\) is already 96° and a triangle cannot contain two obtuse angles. Whenever the given angle is the obtuse one, the angle found by the sine rule must be acute.
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