Sine Rule

The Sine Rule is useful for finding missing angle in non-right-angled triangles when you know one angle and 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. $$ \theta \approx 20.8^\circ \quad \text{(rounded to 1 decimal places)} $$

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