A student begins a proof of the law of cosines. His work is shown.

The next step in the proof is to write an expression for the length of side AZ in terms of x, z, and Zy, z, and Xy, z, and Y.

Then the student should write an expression for the length of side AY in terms of z and Xx and Zz and Y.

Next he can use the Pythagorean theoremtangent functionlaw of sines to relate *x*, *y*, *z*, and XZY.

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To prove the **law of cosines**, we typically start with a triangle and place it in a coordinate system to simplify the derivation. Let’s outline the steps the student is likely following, leading up to the use of the Law of Cosines.

### 1. Define the triangle and label points:
Let’s say we have triangle \( \triangle XYZ \) where:
- \( X \), \( Y \), and \( Z \) represent the angles.
- The sides opposite the angles \( X \), \( Y \), and \( Z \) are \( x \), \( y \), and \( z \) respectively.

For simplicity, let’s place point \( Z \) at the origin of a coordinate system, so \( Z(0, 0) \). Then we can label:
- \( X(x, 0) \), which means that the side \( ZY \) is along the x-axis with length \( x \).
- \( Y \) will be at some coordinates \( (a, b) \), and we will figure out these based on the known side lengths and the angle \( Z \).

### 2. Step 1: Write an expression for side AZ (or XZ):
We are asked to find an expression for the length of side \( AZ \) (or \( XZ \)) in terms of \( x \), \( z \), and the angle between them. This will just be the length of side \( z \) as we already defined it. So:

\[
AZ = z
\]

### 3. Step 2: Write an expression for side AY (or XY):
We need to write the expression for side \( AY \) in terms of \( z \) and the angles. Here’s the next crucial step, as the angle \( Z \) between side \( AY \) and side \( AZ \) is used. 

From trigonometry, using the cosine rule:

\[
AY = \sqrt{z^2 + x^2 - 2zx\cos(Y)}
\]

where \( \cos(Y) \) is the cosine of the angle \( Y \).

### 4. Step 3: Use the Law of Cosines:
We now apply the **law of cosines** to relate the sides \( x \), \( y \), \( z \) and the angles. The law of cosines states that:

\[
z^2 = x^2 + y^2 - 2xy \cos(Z)
\]

This relates the three sides and the cosine of the angle between them, which is exactly what we need to proceed with the proof.

---

### Follow-up Questions:
1. What specific values would you need to plug into the Law of Cosines for an actual problem?
2. How does the Pythagorean Theorem relate to the Law of Cosines when the angle is 90°?
3. Can you derive the Law of Cosines using vector geometry instead of coordinate geometry?
4. How would you use trigonometric identities to simplify expressions in a Law of Cosines proof?
5. Can the Law of Cosines be used to solve for angles instead of sides?

### Tip:
Remember, the Law of Cosines is especially useful when you know two sides and the angle between them or when you know all three sides and need to find an angle!

A student begins a proof of the law of cosines. His work is shown. The next step in the proof is to write an expression for the length of side AZ in terms of x, z, and Zy, z, and Xy, z, and Y. Then the student should write an expression for the length of side AY in terms of z and Xx and Zz and Y. Next he can use the Pythagorean theorem, tangent function, law of sines to relate x, y, z, and XZY.

Explore a detailed step-by-step guide on proving the Law of Cosines. Learn how to express the lengths of sides AZ and AY in terms of known variables and angles. Understand how to apply the Law of Cosines, Pythagorean Theorem, and Law of Sines to relate the sides and angles of a triangle. Suitable for high school students in Grades 9-12.

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