Angle of Elevation and Depression

1Summary 重點摘要

Learn two practical angle terms, then you can solve distance and height measurement problems.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Why learn these two anglesEveryday life is full of lengths we cannot measure directly: you cannot climb a tree to measure its height, nor walk across a river with a tape. Trigonometry solves such problems indirectly: measure one angle and one known horizontal distance, and the height or distance can be calculated. Angles of elevation and depression are the entry tools of indirect measurement, widely used in engineering, navigation and surveying.
2. Step 2: What is angle of elevationThe angle of elevation is the angle between the horizontal line and the line of sight when looking upward. The observer first looks along the horizontal, then lifts the eyes to the object above; the angle between the two lines is the angle of elevation. For example, standing on the ground and looking up at the top of a building forms one. When drawing, remember it is always measured from the horizontal line, not from the ground or the vertical.
3. Step 3: What is angle of depressionThe angle of depression is the angle between the horizontal line and the line of sight when looking downward. From a high position, such as a rooftop, the observer looks along the horizontal and then lowers the eyes to a car on the ground; that angle is the angle of depression. It is also measured from the horizontal, only downward. If the object is above eye level it is elevation; if below, it is depression — always draw the horizontal line first.
4. Step 4: Relation between themElevation and depression are always equal. The two horizontal lines are parallel, and the line of sight between the observer and the target is one straight line, so the elevation at one end and the depression at the other form a pair of alternate angles. Your angle looking up at someone on the roof equals their angle looking down at you. In problems, once one angle is known, the other can be written down at once.
5. Step 5: How to solve problemsProblem-solving follows a fixed routine. First draw the right triangle and mark the horizontal, the line of sight and the angle; then identify the opposite, adjacent and hypotenuse, and list what is known and what is wanted. Next, choose the right trig ratio — tan is usual when a horizontal distance is given and a height is wanted, because tanθ = opposite ÷ adjacent. Finally, substitute, calculate and check that the answer is sensible.
6. Step 6: ExampleExample: standing 10 m from a tree, the angle of elevation to the top is 45°; find the height. Draw the right triangle: adjacent = 10 m, angle = 45°, opposite = tree height. Since the adjacent is known and the opposite is wanted, use tan: tan45° = opposite ÷ adjacent = height ÷ 10, and tan45° = 1, so the height = 10 × 1 = 10 m. Check: at 45° the opposite equals the adjacent, so 10 m against 10 m is reasonable.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Elevation: looking up
• Depression: looking down
• Two angles always equal
• Draw right triangle first before calculating
• Usually use tan to calculate