write sine and cosine rules and in which case they would be used

### Sine Rule (Law of Sines):
The sine rule relates the ratios of the sides of a triangle to the sines of its angles.

For any triangle with angles \( A \), \( B \), and \( C \), and opposite sides \( a \), \( b \), and \( c \) respectively:
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]

This rule is typically used in the following cases:
1. **When you know two angles and one side (AAS or ASA)**: You can use the sine rule to find the missing sides.
2. **When you know two sides and one non-included angle (SSA)**: You can use the sine rule to find the missing angles.

### Cosine Rule (Law of Cosines):
The cosine rule relates the sides of a triangle to the cosine of one of its angles.

For any triangle with angles \( A \), \( B \), and \( C \), and opposite sides \( a \), \( b \), and \( c \) respectively:
\[
a^2 = b^2 + c^2 - 2bc \cdot \cos A
\]
\[
b^2 = a^2 + c^2 - 2ac \cdot \cos B
\]
\[
c^2 = a^2 + b^2 - 2ab \cdot \cos C
\]

This rule is typically used in the following cases:
1. **When you know all three sides (SSS)**: You can use the cosine rule to find the angles.
2. **When you know two sides and the included angle (SAS)**: You can use the cosine rule to find the third side.

### Summary of Use Cases:
- **Sine Rule**: Use when you have AAS, ASA, or SSA information.
- **Cosine Rule**: Use when you have SSS or SAS information.

Would you like further details on these rules or have any specific questions?

#### Related Questions:
1. How can the sine rule be applied to non-right triangles?
2. What happens if the angles given don't sum up to 180° in the sine rule?
3. How do you derive the cosine rule?
4. Can the sine or cosine rule be used for right triangles?
5. How do the sine and cosine rules apply to obtuse triangles?

**Tip**: The sine rule can sometimes give two possible triangle configurations in SSA cases, known as the ambiguous case. Be cautious!

Explore the sine and cosine rules, key trigonometric concepts that relate triangle angles and sides. Learn when to apply the sine rule (AAS, ASA, SSA) and the cosine rule (SSS, SAS) in solving triangle problems. Suitable for students in Grades 9-12.

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