Towers of Reflection?

📌 CONCEPT: In trigonometry, the angles of elevation to the top of an object from a point on the ground (or in this case, a boat in the river) can be used to calculate the height of the object using trigonometric ratios.

📐 RULE / FORMULA: We can use the tangent function to relate the angles of elevation to the heights of the towers: tan(angle) = opposite side (height of tower) / adjacent side (distance from boat to tower).

💡 WORKED EXAMPLE: Let's assume the distance from the boat to each tower is x. Using the tangent function, we can write: tan(30°) = h1 / x and tan(60°) = h2 / x. Given that tan(30°) = 1/√3 and tan(60°) = √3, we can solve for the heights h1 and h2. By cross-multiplication, we get: h1 = x√3 and h2 = 3x. Taking the ratio of the heights, we get: h1/h2 = √3/3.

⚠️ COMMON MISTAKE: Students often forget to consider the relationships between the angles of elevation and the heights of the towers, or they may confuse the tangent function with the sine or cosine functions.