Choosing Between Sine and Cosine Rules

Worked example

1. 1

   $B{C}^2=7^2+9^2-2(7)(9)\cos 65^{\circ }$ — In triangle $ABC$ we know two sides (7 and 9) and the angle between them, and want the side opposite that angle. There is no side-and-facing-angle pair, so use the cosine rule.

2. 2

   $B{C}^2=130-126\cos 65^{\circ }$ — Square the sides: 49 + 81 = 130. Multiply 2 × 7 × 9 = 126.

3. 3

   $B{C}^2=130-53.25=76.75$ — Work out $126\cos 65^{\circ }=53.25$ first, then subtract from 130.

4. 4

   $BC=\sqrt{76.75}=8.761$ — Square root to get $BC$. Keep 4 figures: it is the shared side, used again next.

5. 5

   $\frac{CD}{\sin 40^{\circ }}=\frac{8.761}{\sin 58^{\circ }}$ — Now triangle $BCD$ has a pair: $BC$ faces the 58° angle. $CD$ faces the 40° angle, so use the sine rule.

6. 6

   $CD=\frac{8.761\times \sin 40^{\circ }}{\sin 58^{\circ }}$ — Multiply both sides by $\sin 40^{\circ }$ to get $CD$ on its own.

7. 7

   $CD=6.64\text{ cm}$ — Work it out and round to 3 significant figures, only now at the end.

Worked example

1. 1

   $\frac{1}{2}\times 8\times 11\times \sin A=30$ — Area = ½ab sin C, with sides 8 and 11 around angle $A$. The area is given and $A$ is unknown.

2. 2

   $44\sin A=30$ — Half of 8 × 11 is 44.

3. 3

   $\sin A=\frac{30}{44}=0.6818$ — Divide both sides by 44.

4. 4

   $A=42.99^{\circ }$ — Inverse sine. The question says $A$ is acute, so this is the angle we want.

5. 5

   $B{C}^2=8^2+11^2-2(8)(11)\cos A$ — Now two sides and the angle between them are known, so use the cosine rule for the third side.

6. 6

   $B{C}^2=185-176\cos 42.99^{\circ }$ — Square the sides: 64 + 121 = 185. Multiply 2 × 8 × 11 = 176.

7. 7

   $B{C}^2=185-128.75=56.25$ — Work out $176\cos 42.99^{\circ }=128.75$, then subtract from 185. Use the full calculator value of $A$, not a rounded one.

8. 8

   $BC=\sqrt{56.25}=7.50\text{ cm}$ — Square root, to 3 significant figures.

Worked example

1. 1

   $\frac{\sin x}{6.2}=\frac{\sin 110^{\circ }}{9.5}$ — The side 9.5 faces the 110° angle, so there is a pair: use the sine rule. $x$ faces 6.2, with the unknown angle on top.

2. 2

   $\sin x=\frac{6.2\times \sin 110^{\circ }}{9.5}$ — Multiply both sides by 6.2.

3. 3

   $\sin x=0.6133$ — Work it out: 6.2 × 0.9397 ÷ 9.5.

4. 4

   $x={\sin }^{-1}(0.6133)=37.8^{\circ }$ — Inverse sine, to 1 decimal place.

Worked example

1. 1

   $\cos C=\frac{5^2+8^2-11^2}{2(5)(8)}$ — The largest angle faces the longest side, 11. In the rearranged cosine rule, the side facing the angle you want is the one that is subtracted.

2. 2

   $\cos C=\frac{25+64-121}{80}$ — Square each side. Multiply 2 × 5 × 8 = 80.

3. 3

   $\cos C=\frac{-32}{80}=-0.4$ — 25 + 64 = 89, and 89 − 121 = −32. Keep the minus sign.

4. 4

   $C={\cos }^{-1}(-0.4)=113.6^{\circ }$ — Inverse cosine of a negative number gives an obtuse angle. That fits: the longest side faces the biggest angle.