The captain of a boat at sea, at point Q, notices a lighthouse PM directly north of his position - NSC Mathematics - Question 7 - 2018 - Paper 2

Using the right triangle formed by points Q, M, and P, we can apply trigonometric ratios. From the figure:

$QM = 12x \cos(\beta)$ Substituting QM from part 7.1 gives:

$\frac{x}{\tan(\theta)} = 12x \cos(\beta)$ Dividing both sides by x (assuming x ≠ 0):

$\frac{1}{\tan(\theta)} = 12 \cos(\beta)$ Thus, we have:

$\tan(\theta) = \frac{\cos(\beta)}{12}$ This completes the proof.

Using the relationship found above:
Since QM = 60 metres, we substitute into our equation:

$60 = \frac{x}{\tan(\theta)} \implies x = 60 \tan(\theta)$ We also know from step 7.2 that:

$\tan(40°) = \frac{\cos(40°)}{12}$ Calculating:

$x = 60 \tan(40°) \approx 60 \cdot 0.8391 \approx 50.35$.
Rounding to the nearest metre, the height of the lighthouse, x, is approximately 50 metres.