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GB-33-04-001 · Sine rule

Triangle ABC has angle A = 57°, angle B = 91° and side BC = 23 cm. Calculate AC to 1 decimal place.

Sine Rule ("Sine Sausages!" — Finding Length) Booklet p. 36
Method
1) Circle two opposite side–angle pairs
2) Relabel the unknown length a
3) Use a / sin A = b / sin B
4) Substitute the paired values and solve
Worked Example (GB-33-04-001)
1)
Circle the two opposite side–angle pairs: unknown AC is opposite 91∘91^\circ; known BC = 2323 is opposite 57∘57^\circ.
2)
Relabel the unknown length as a, as in the booklet: a=AC, A=91∘, b=23, B=57∘a=AC,\ A=91^\circ,\ b=23,\ B=57^\circ. These are local sine-rule labels.
3)
asin⁡A=bsin⁡B\frac{a}{\sin A}=\frac{b}{\sin B}
4)
Substitute: asin⁡91∘=23sin⁡57∘\frac{a}{\sin91^\circ}=\frac{23}{\sin57^\circ}.
5)
Multiply by sin⁡91∘\sin91^\circ: a=23sin⁡91∘sin⁡57∘=27.4a=\frac{23\sin91^\circ}{\sin57^\circ}=27.4 cm, so AC = 27.427.4 cm.
Ans = 27.4 cm