Sine Rule

The Sine Rule is useful for finding missing sides in non-right-angled triangles when you know one angle and its opposite side.

1. Identify Given Information
   
   • You know one angle (30°) and its opposite side (4 cm).
   • You also know another angle (70°) and need to find its opposite side.
2. Apply the Sine Rule
   
   • Using the Sine Rule formula: $$\frac{\sin A}{a}=\frac{\sin B}{b}$$​
   • Substitute the known values: $$\frac{\sin ({30}^{\circ })}{4\text{ cm}}=\frac{\sin ({70}^{\circ })}{?}$$
3. Solve for the Missing Side
   
   1. Rearranging the equation: $$?=\frac{4\text{ cm}\times \sin ({70}^{\circ })}{\sin ({30}^{\circ })}$$
   2. Now, calculate the value to find the missing side. $$?\approx 7.52\text{ cm}(\text{rounded to 2 decimal places})$$

🍀 Exam Tip: When using the Sine Rule, always set up the equation carefully by matching each angle to its opposite side.

1. Identify Given Information
   • You know one angle ${135}^{\circ }$ and its opposite side $10\text{ cm}$.
   • You also know another side $5\text{ cm}$ and need to find its opposite angle $\theta$.
2. Apply the Sine Rule
   • Using the Sine Rule formula:
     $$\frac{\sin A}{a}=\frac{\sin B}{b}$$
   • Substituting the known values: $$\frac{\sin ({135}^{\circ })}{10\text{ cm}}=\frac{\sin (\theta )}{5\text{ cm}}$$
3. Solve for the Missing Angle
   • Rearranging the equation:
     $$\sin (\theta )=\frac{5\text{ cm}\times \sin ({135}^{\circ })}{10\text{ cm}}$$
   • Now, since $\sin ({135}^{\circ })\approx 0.71$, $$\sin (\theta )\approx \frac{5\times 0.71}{10}\approx 0.355$$
   • Now, use the inverse sine function to find the missing angle. $$\theta \approx {\sin }^{-1}(0.355)$$
   • On your calculator, press sin⁻¹, then enter 0.3555, and press = to get the angle. $$\theta \approx {20.78}^{\circ }\text{(rounded to 2 decimal places)}$$

🍀 Exam Tip: When using the Sine Rule to find angles, always apply ${\sin }^{-1}$ (inverse sine) to get the angle.