In \( \Delta \mathrm{POQ} \), use the cosine rule to determine the distance of PQ (in terms of \( \alpha \) and \( \beta \) ) F quate the PQ in 23 and the PQ in 2.4 , then make \( \cos (\alpha-\beta) \) the subject of the formula.

To find the distance \( PQ \) in terms of \( \alpha \) and \( \beta \) using the cosine rule, we can express it as follows: \[ PQ^2 = PO^2 + OQ^2 - 2 \cdot PO \cdot OQ \cdot \cos(\angle POQ) \] Assuming \( PO = 23 \) and \( OQ = 2.4 \), substituting these values gives: \[ PQ^2 = 23^2 + 2.4^2 - 2 \cdot 23 \cdot 2.4 \cdot \cos(\angle POQ) \] This can be further simplified to find \( PQ \) once you compute the values and rearrange for \( \cos(\alpha - \beta) \). Now, rearranging to make \( \cos(\alpha - \beta) \) the subject results in: \[ \cos(\angle POQ) = \frac{23^2 + 2.4^2 - PQ^2}{2 \cdot 23 \cdot 2.4} \] Using the identity \( \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \), we can express it in terms of the angles or apply a specific numerical example with definite angles \( \alpha \) and \( \beta \) for further simplification. For real-world applications, knowing the cosine rule is invaluable. Whether calculating distances in navigation, construction, or even video game design, this tool helps us solve trigonometric problems involving non-right triangles, ensuring accuracy in various fields. When utilizing the cosine rule, a common mistake is forgetting to square the lengths when applying the formula. Always check if you've correctly squared the lengths before subtracting, or you might end up with incorrect values, leading to confused calculations and frustration!