Maths

• What are the Sine and Cosine Rules used for?


  They are used to solve triangles.
• How should the sides of a triangle be labeled in relation to its angles?


  Side 'a' is opposite angle A, side 'b' is opposite angle B, and side 'c' is opposite angle C.
• What are the three formulas to learn for the Sine and Cosine Rules?

  1. Sine Rule: $\frac{a}{\sin A} =$$\frac{b}{\sin B} =$$\frac{c}{\sin C}$
  2. Cosine Rule: $a^2 =$$b^2 +$$c^2 - 2bc \cos A$
  3. Area of Triangle: $Area =$$\frac{1}{2}ab \sin C$
• What does the Sine Rule relate?


  The Sine Rule relates the lengths of the sides of a triangle to the sines of its angles.
• How do you calculate the area of a triangle when you know two sides and the angle between them?


  Use the formula: Area = $\frac{1}{2}ab \sin C$.
• Given triangle XYZ with sides XZ = 18 cm, YZ = 13 cm, and angle XZY = 58°, how do you find the area?


  Area = $\frac{1}{2} \times 18 \times 13 \times \sin 58$.
• What is the formula for the area of a triangle when two sides and the included angle are known?

  Area = $\frac{1}{2}ab \sin C$.
• If two angles and any side of a triangle are given, which rule is needed?

  The Sine Rule is needed.
• How do you find the length of side AB when given two angles and one side?

  Use the Sine Rule: $\frac{b}{\sin B} =$$\frac{c}{\sin C}$.
• What is the first step to find angle ABC when given two sides and an angle not enclosed by them?


  Put the numbers into the sine calculation.
• How do you find sin B when given two sides and angle A?


  Rearrange to find sin B: $sin B =$$\frac{b \sin A}{a}$.
• What is the formula for the Cosine Rule?


  The formula is $a^2 =$$b^2 +$$c^2 - 2bc \cos A$.
• How do you find the length of side CB when given two sides and the angle enclosed by them?


  Use the Cosine Rule: $a^2 =$$b^2 +$$c^2 - 2bc \cos A$.
• What should you do if you encounter a triangle that isn't labeled ABC?

  Relate it yourself to match the Sine and Cosine Rules.
• How do you find angle CAB when all three sides are given but no angles?


  Use the Cosine Rule: $\cos A =$$\frac{b^2 + c^2 - a^2}{2bc}$.
• What is the first step to find angle A using the Cosine Rule?


  Put in the numbers into the formula: $\cos A =$$\frac{b^2 + c^2 - a^2}{2bc}$.
• How do you find the inverse to determine angle A?


  Use $A =$$\cos^{-1}(value)$ after calculating cos A.
• What is the final answer for angle A when calculated using the Cosine Rule?


  Angle A = 83.3° (1 d.p.).
• What are the different scenarios for using the Sine and Cosine Rules?

  1. Two angles and any side — Sine Rule needed.
  2. Two sides and an angle not enclosed — Sine Rule needed.
  3. Two sides and the angle enclosed — Cosine Rule needed.
  4. All three sides given but no angles — Cosine Rule needed.