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Let's break down and solve the problem based on what is shown in the image.

### a) Show that the cosine rule can be rearranged into the form:
\[
\cos A = \frac{b^2 + c^2 - a^2}{2bc}
\]
The cosine rule is given as:
\[
a^2 = b^2 + c^2 - 2bc \cdot \cos A
\]
To rearrange it for \(\cos A\):

1. Start with the given equation:
   \[
   a^2 = b^2 + c^2 - 2bc \cdot \cos A
   \]
2. Subtract \(b^2 + c^2\) from both sides:
   \[
   a^2 - b^2 - c^2 = -2bc \cdot \cos A
   \]
3. Divide both sides by \(-2bc\):
   \[
   \cos A = \frac{b^2 + c^2 - a^2}{2bc}
   \]
Thus, we have shown that the cosine rule can indeed be rearranged into the desired form.

---

### b) Finding the size of angle \(\theta\)

Now, using the cosine rule, we can find \(\theta\) in the triangle given the sides:
- \(a = 23 \, \text{mm}\)
- \(b = 15 \, \text{mm}\)
- \(c = 12 \, \text{mm}\)

We apply the cosine rule:
\[
\cos \theta = \frac{b^2 + c^2 - a^2}{2bc}
\]
Substitute the given values:
\[
\cos \theta = \frac{15^2 + 12^2 - 23^2}{2 \cdot 15 \cdot 12}
\]
\[
\cos \theta = \frac{225 + 144 - 529}{2 \cdot 15 \cdot 12}
\]
\[
\cos \theta = \frac{-160}{360}
\]
\[
\cos \theta = -0.4444
\]
Now, calculate \(\theta\):
\[
\theta = \cos^{-1}(-0.4444)
\]
Using a calculator:
\[
\theta \approx 116.4^\circ
\]
Therefore, the size of angle \(\theta\) is approximately \(116^\circ\) to the nearest degree.

---

Do you need more details or further clarifications on any step?

Here are some related questions you might find interesting:
1. How would the cosine rule change for an obtuse triangle?
2. Can the cosine rule be applied to non-triangular shapes like quadrilaterals?
3. How can the cosine rule be used to find unknown sides of a triangle?
4. What is the difference between the cosine rule and the sine rule?
5. How can the area of a triangle be calculated using trigonometry?

**Tip**: The cosine rule is especially useful for non-right-angled triangles, helping to find both angles and sides when different sets of information are known.

a) Show that the cosine rule shown below can be rearranged to give cos(A) = (b^2 + c^2 - a^2) / (2bc). b) What is the size of angle θ in the triangle below? Give your answer to the nearest degree.

Learn how to rearrange the cosine rule and find the size of an angle in a triangle using given side lengths. This example covers deriving the formula for cos(A) and calculating angle θ using the cosine rule. Suitable for high school students in Grades 9-12.

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