Sine Rule

• What is the Sine Rule formula for sides?

  \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)
• What does the Sine Rule help to find?

  Missing sides or angles in triangles
• If \( a = 12 \text{ cm} \), \( A = 50^\circ \), and \( B = 87^\circ \), how do you find side \( b \)?

  Use \( b = \frac{\sin B \cdot a}{\sin A} \)
• What is the calculated length of side \( b \) when \( a = 12 \text{ cm} \), \( A = 50^\circ \), and \( B = 87^\circ \)?

  15.64 cm
• How do you find side \( R \) using the Sine Rule?

  • Given: \( S = 84 \text{ cm} \), \( R = \text{unknown} \)
  • Use: \( \frac{\sin R}{\sin S} = \frac{R}{S} \)
  • Calculate: \( R = \frac{\sin(63^\circ) \cdot 17}{\sin(84^\circ)} \)
  • Result: 15.2 cm
• What is the formula to find side \( T \) using the Sine Rule?


  \( \frac{\sin S}{\sin T} = \frac{S}{T} \)
• If \( S = 124 \text{ cm} \) and \( T = \text{unknown} \), how do you find \( T \)?

  Use \( T = \frac{\sin S \cdot t}{\sin T} \)
• What is the calculated length of side \( T \) when \( S = 124 \text{ cm} \) and \( \sin(59^\circ) \)?

  4.85 cm
• How do you find side \( U \) using the Sine Rule?

  • Given: \( 25 \text{ cm} \), \( \sin(40^\circ) \)
  • Use: \( \frac{\sin U}{\sin 123^\circ} = \frac{25}{U} \)
  • Calculate: \( U = \frac{\sin(40^\circ) \cdot 25}{\sin(123^\circ)} \)
  • Result: 19.2 cm
• What is the relationship between angles \( D \) and \( F \) in the Sine Rule?

  \( \frac{\sin D}{\sin F} = \frac{d}{f} \)
• If \( d = 10 \text{ cm} \) and \( F = 94^\circ \), how do you find side \( f \)?

  Use \( f = \frac{\sin F \cdot d}{\sin D} \)
• What is the calculated length of side \( f \) when \( d = 10 \text{ cm} \) and \( F = 94^\circ \)?


  15.2 cm
• How do you find side \( G \) using the Sine Rule?

  • Given: \( e = 8 \text{ cm} \), \( \sin(25^\circ) \)
  • Use: \( \frac{\sin G}{\sin E} = \frac{g}{e} \)
  • Calculate: \( G = \frac{\sin(25^\circ) \cdot 8}{\sin(93^\circ)} \)
  • Result: 3.39 cm
• What is the formula for the cosine of angle A in the sine rule?

  $\cos A =$$\frac{b^2 + c^2 - a^2}{2bc}$
• What is the formula to find the length a in the sine rule?

  $a =$$\sqrt{b^2 + c^2 - 2b\cos A}$
• What does the angle represent in the sine rule?

  The angle is in the middle of the formula
• How do you find angle ∠MNP using cosine?

  Use $\cos N =$$\frac{m^2 + p^2 - n^2}{2mp}$
• What is the calculation for $32^2 =$$17^2 - 45^2$?


  It simplifies to $32^2 =$$289 - 2025$
• What is the value of $\cos N$ calculated in the example?


  $\cos N =$$-0.91$
• How do you find angle N from $\cos N =$$-0.91$?


  Use $N =$$\cos^{-1}(-0.91)$
• What is the approximate value of angle N?

  $N \approx 155.622^\circ$
• How do you find angle M using the cosine rule?

  Use $\cos M =$$\frac{17^2 + 48^2 - 32^2}{2 \times 17 \times 48}$
• What is the calculated value of $\cos M$?


  $\cos M \approx 0.96$
• How do you find angle M from $\cos M \approx 0.96$?


  Use $M =$$\cos^{-1}(0.96)$
• What is the approximate value of angle M?

  $M \approx 16.3^\circ$
• How do you find angle P using the cosine rule?

  Use $\cos P =$$\frac{32^2 + 48^2 - 17^2}{2 \times 32 \times 48}$
• What is the calculated value of $\cos P$?


  $\cos P \approx 0.989$
• How do you find angle P from $\cos P \approx 0.989$?


  Use $P =$$\cos^{-1}(0.989)$
• What is the approximate value of angle P?

  $P \approx 8.1^\circ$