Sine rule questions with answers

Questions

1. Question 1Find a side

   In triangle $ABC$, angle $BAC = 32^\circ$, angle $ABC = 44^\circ$ and $BC = 6$ cm. Find the length of $AC$, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: Which rule links the sides and angles you know with the one you want?$\text{Sine rule}$
   2. Step 2: Pair each side with the angle opposite it$BC = 6$ is opposite angle $A = 32^\circ$$AC = x$ is opposite angle $B = 44^\circ$
   3. Step 3: Write the rule with the values$\dfrac{a}{\sin A} = \dfrac{b}{\sin B}$$\dfrac{6}{\sin 32^\circ} = \dfrac{x}{\sin 44^\circ}$
   4. Step 4: Make x the subject$x = \dfrac{6 \sin 44^\circ}{\sin 32^\circ}$$x = 7.8652\ldots$$\phantom{x} = 7.87$ cm (3 s.f.)
2. Question 2Find a side: third angle first

   In triangle $ABC$, angle $BAC = 73^\circ$, angle $ACB = 38^\circ$ and $BC = 10$ cm. Find the length of $AC$, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: Find the angle opposite the side you wantAngle $ABC = 180^\circ - 73^\circ - 38^\circ$$ABC = 69^\circ$
   2. Step 2: Which rule links the sides and angles you know with the one you want?$\text{Sine rule}$
   3. Step 3: Pair each side with the angle opposite it$BC = 10$ is opposite angle $A = 73^\circ$$AC = x$ is opposite angle $B = 69^\circ$
   4. Step 4: Write the rule with the values$\dfrac{a}{\sin A} = \dfrac{b}{\sin B}$$\dfrac{10}{\sin 73^\circ} = \dfrac{x}{\sin 69^\circ}$
   5. Step 5: Make x the subject$x = \dfrac{10 \sin 69^\circ}{\sin 73^\circ}$$x = 9.7623\ldots$$\phantom{x} = 9.76$ cm (3 s.f.)
3. Question 3Find an acute angle

   In triangle $ABC$, $BC = 13$ cm, $AC = 12$ cm and angle $BAC = 38^\circ$. Find angle $ABC$, correct to 1 decimal place.

   Show the worked solution

   1. Step 1: Which rule links the sides and angles you know with the one you want?$\text{Sine rule}$
   2. Step 2: Pair each side with the angle opposite it$BC = 13$ is opposite angle $A = 38^\circ$$AC = 12$ is opposite angle $B = \theta$
   3. Step 3: Write the rule with the angles on top, with the values$\dfrac{\sin B}{b} = \dfrac{\sin A}{a}$$\dfrac{\sin \theta}{12} = \dfrac{\sin 38^\circ}{13}$
   4. Step 4: Make sin θ the subject$\sin \theta = \dfrac{12 \sin 38^\circ}{13}$$\phantom{\sin \theta} = 0.5683\ldots$
   5. Step 5: Use sin⁻¹$\theta = \sin^{-1}(0.5683\ldots)$$\theta = 34.6319\ldots^\circ$$\phantom{\theta} = 34.6^\circ$ (1 d.p.)
4. Question 4Find an obtuse angle

   In triangle $ABC$, $BC = 9$ cm, $AC = 12$ cm and angle $BAC = 38^\circ$. Given that angle $ABC$ is obtuse, find angle $ABC$, correct to 1 decimal place.

   Show the worked solution

   1. Step 1: Which rule links the sides and angles you know with the one you want?$\text{Sine rule}$
   2. Step 2: Pair each side with the angle opposite it$BC = 9$ is opposite angle $A = 38^\circ$$AC = 12$ is opposite angle $B = \theta$
   3. Step 3: Write the rule with the angles on top, with the values$\dfrac{\sin B}{b} = \dfrac{\sin A}{a}$$\dfrac{\sin \theta}{12} = \dfrac{\sin 38^\circ}{9}$
   4. Step 4: Make sin θ the subject$\sin \theta = \dfrac{12 \sin 38^\circ}{9}$$\phantom{\sin \theta} = 0.8208\ldots$
   5. Step 5: The calculator gives the acute angle; θ is obtuse, so use sin θ = sin(180° − θ)Acute angle $= \sin^{-1}(0.8208\ldots)$$= 55.1731\ldots^\circ$$\sin \theta = \sin(180^\circ - 55.1731\ldots^\circ)$$\theta = 180^\circ - 55.1731\ldots^\circ$$\phantom{\theta} = 124.8^\circ$ (1 d.p.)