The Sine Rule (or Law of Sines) is a fundamental relationship in trigonometry that applies to any triangle, not just right-angled ones. It states that in any triangle \(ABC\):

\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]

where \(a\), \(b\), and \(c\) are the lengths of the sides opposite the angles \(A\), \(B\), and \(C\), respectively.

### Proof of the Sine Rule:

Consider any triangle \(ABC\) with sides \(a\), \(b\), and \(c\) opposite to angles \(A\), \(B\), and \(C\) respectively.

1. **Draw the height from \(C\) to the base \(AB\)**:  
   Let the height \(h\) from vertex \(C\) drop perpendicularly onto side \(AB\), splitting it into two segments, \(p\) and \(q\), where \(p + q = c\).

2. **Express the height \(h\) using the sine of angles**:  
   - From triangle \(ACD\) (where \(D\) is the foot of the perpendicular), we have:
     \[
     h = b \sin A
     \]
   - From triangle \(BCD\), we have:
     \[
     h = a \sin B
     \]

3. **Equate the two expressions for \(h\)**:
   Since both expressions represent the height \(h\), we equate them:
   \[
   b \sin A = a \sin B
   \]

4. **Rearrange to get a part of the Sine Rule**:
   \[
   \frac{a}{\sin A} = \frac{b}{\sin B}
   \]

5. **Repeat the process for another height**:
   Now draw the height from vertex \(A\) to side \(BC\). This leads to:
   \[
   c \sin B = b \sin C
   \]
   which can be rearranged to:
   \[
   \frac{b}{\sin B} = \frac{c}{\sin C}
   \]

6. **Combine the two results**:
   By combining the two equalities, we get the complete Sine Rule:
   \[
   \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
   \]

This completes the proof of the Sine Rule.

Would you like more details on any of the steps, or do you have any questions?

Here are some follow-up questions to expand on this topic:

1. How does the Sine Rule apply to right-angled triangles specifically?
2. Can the Sine Rule be used to find the area of a triangle?
3. How is the Sine Rule related to the Cosine Rule?
4. In what scenarios might the Sine Rule be more useful than the Cosine Rule?
5. How does the Sine Rule generalize to spherical triangles?

**Tip**: The Sine Rule is especially useful in non-right-angled triangles where direct trigonometric methods for right-angled triangles (like SOH-CAH-TOA) don't apply.

Explore the proof of the Sine Rule, a key concept in trigonometry that applies to any triangle. This step-by-step explanation covers the derivation of the formula \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\), useful for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation