Law of Cosines Cosine Rule Proof

📐 Understanding the Law of Cosines

The Law of Cosines, also known as the Cosine Rule, is a fundamental trigonometric identity that relates the sides and angles of a triangle. It's especially useful for solving triangles when you know either:

• Two sides and the included angle (SAS)
• All three sides (SSS)

The formula is given by:

$$c^2 = a^2 + b^2 - 2ab \cdot cos(C)$$ Where:

• $a$, $b$, and $c$ are the lengths of the sides of the triangle.
• $C$ is the angle opposite side $c$.

📝 Proof of the Cosine Rule

Let's walk through a step-by-step proof to understand why this formula works.

1. Consider a Triangle:

   Start with a triangle ABC, where side $a$ is opposite angle A, side $b$ is opposite angle B, and side $c$ is opposite angle C.

2. Draw an Altitude:

   Draw an altitude (height) from vertex B to side AC. Let's call the point where the altitude meets AC point D. This divides side $b$ into two segments: $x$ and $(b - x)$.

3. Apply the Pythagorean Theorem:

   Now we have two right triangles: ABD and CBD. Apply the Pythagorean theorem to both.

   • In triangle ABD: $h^2 + x^2 = c^2$
   • In triangle CBD: $h^2 + (b - x)^2 = a^2$
4. Express $h^2$:

   From both equations, express $h^2$:

   • $h^2 = c^2 - x^2$
   • $h^2 = a^2 - (b - x)^2$
5. Equate and Simplify:

   Since both expressions equal $h^2$, we can set them equal to each other:

   $c^2 - x^2 = a^2 - (b - x)^2$

   Expand and simplify:

   $c^2 - x^2 = a^2 - (b^2 - 2bx + x^2)$

   $c^2 - x^2 = a^2 - b^2 + 2bx - x^2$

   $c^2 = a^2 - b^2 + 2bx$

6. Relate $x$ to Cosine:

   In right triangle ABD, we can relate $x$ to angle A using cosine:

   $cos(A) = \frac{x}{c}$

   So, $x = c \cdot cos(A)$

7. Substitute $x$:

   Substitute $x$ in the equation from step 5:

   $c^2 = a^2 - b^2 + 2b(c \cdot cos(A))$

8. Rearrange to Cosine Rule:

   Rearrange the equation to isolate $c^2$:

   $c^2 = a^2 + b^2 - 2ab \cdot cos(C)$

   Note: The angle used in the cosine term will be the angle opposite side c. Therefore, replacing angle A with angle C, we have:

   $c^2 = a^2 + b^2 - 2ab \cdot cos(C)$

💡 Example

Let's say we have a triangle with sides $a = 5$, $b = 7$, and angle $C = 60^\circ$. We want to find side $c$.

Using the Law of Cosines:

$c^2 = 5^2 + 7^2 - 2 \cdot 5 \cdot 7 \cdot cos(60^\circ)$

$c^2 = 25 + 49 - 70 \cdot 0.5$

$c^2 = 74 - 35$

$c^2 = 39$

$c = \sqrt{39} \approx 6.24$

🔑 Key Takeaways

• The Law of Cosines is a generalization of the Pythagorean theorem.
• It's applicable to all triangles, not just right triangles.
• It helps in solving triangles when you have SAS or SSS information.