SL 3.3—Angles of elevation and depression, bearings

Trigonometry isn't just an abstract exercise , it's one of the oldest practical tools in human history. Before GPS and smartphones, surveyors, navigators, and architects used angles and trigonometric ratios to measure heights, distances, and directions with remarkable precision.

In this subtopic, you'll learn two key applications:

• Angles of elevation and depression , used to find heights and distances when you can't measure directly
• Bearings , a standardised system for describing directions, essential in navigation and surveying

Both topics draw on the trigonometric ratios (sin, cos, tan) and, for more complex problems, the sine rule and cosine rule.

Angle of Elevation: The angle formed between the horizontal line of sight and a line of sight directed upward toward an object. It is always measured from the horizontal up to the line of sight.

Whenever you tilt your head upward to look at the top of a building, a mountain peak, or an aeroplane, the angle your line of sight makes with the horizontal is the angle of elevation.

Setting up the trigonometry:
In a right-angled triangle formed by the horizontal distance, the vertical height, and the line of sight:

• The opposite side is the vertical height of the object
• The adjacent side is the horizontal distance from the observer to the base of the object
• The hypotenuse is the line of sight

This means $\tan (\theta )=\frac{\text{height}}{\text{horizontal distance}}$ is often the most direct relationship to use.