Moderate -0.5 This is a straightforward application of the sine rule to find one side given two angles and one side. The angles are special angles (45° and 60°) allowing exact values, but the calculation is routine with no problem-solving required beyond recognizing which rule to apply.
1 In the triangle \(A B C , A B = 12 \mathrm {~cm}\), angle \(B A C = 60 ^ { \circ }\) and angle \(A C B = 45 ^ { \circ }\). Find the exact length of \(B C\).
Use of sine rule \(\frac{12}{\sin 45} = \frac{x}{\sin 60}\)
M1
Used correctly in their triangle \(ABC\)
\(\sin 60 = \frac{\sqrt{3}}{2}\) and \(\sin 45 = \frac{1}{\sqrt{2}}\)
B1
Both of these correct
\(\rightarrow BC = 6\sqrt{3}\sqrt{2}\) or \(6\sqrt{6}\) or \(\sqrt{216}\)
A1
Co – must be in surd form. [3]
Use of sine rule $\frac{12}{\sin 45} = \frac{x}{\sin 60}$ | M1 | Used correctly in their triangle $ABC$
$\sin 60 = \frac{\sqrt{3}}{2}$ and $\sin 45 = \frac{1}{\sqrt{2}}$ | B1 | Both of these correct
$\rightarrow BC = 6\sqrt{3}\sqrt{2}$ or $6\sqrt{6}$ or $\sqrt{216}$ | A1 | Co – must be in surd form. [3]
1 In the triangle $A B C , A B = 12 \mathrm {~cm}$, angle $B A C = 60 ^ { \circ }$ and angle $A C B = 45 ^ { \circ }$. Find the exact length of $B C$.
\hfill \mbox{\textit{CAIE P1 2008 Q1 [3]}}