Trigonometric Applications

1Summary 重點摘要

Learn angle of elevation, depression, bearing, solve real world problems.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Draw a diagram and turn the words into a right triangleAlways start a trigonometry application question by drawing a right-angled triangle. Sort the information in the wording onto the triangle: mark every known side, every known angle, and the side or angle you need to find. A diagram turns a long word problem into a clear picture of the spatial relationships. The classic mistake is applying a formula before identifying the opposite and adjacent sides, which ruins the whole question.
2. Step 2: Angle of elevation — measured up from the horizontalThe angle of elevation is the angle between the horizontal line and the line of sight when you look UP at an object. Standing on the ground and looking at the top of a tree, the angle your sight line makes above the horizontal is the angle of elevation; an elevation of 30° means the sight line sits 30° above the horizontal. Beware: it is measured from the horizontal, never from the vertical, so draw a dashed horizontal line first and measure the angle from it.
3. Step 3: Angle of depression — measured down from the horizontalThe angle of depression is the angle below the horizontal when you look DOWN from a height. The two horizontal lines are parallel, so the depression from the top and the elevation from the bottom are alternate angles and equal: an elevation of 25° means the corresponding depression is also 25°. This parallel-line fact is the bridge in most depression problems. A classic error is measuring the depression from the vertical instead of the horizontal — once the reference is wrong, every angle in the triangle follows.
4. Step 4: Bearings — clockwise from north, three digitsBearings are measured clockwise from due north and must be written as three digits: north 000°, east 090°, south 180°, west 270°, and northeast is 045°, not 45°. Draw a compass cross at every point mentioned in the question, mark the north-south line and then the angle. Remember that the bearing of B from A and the bearing of A from B differ by 180°; sketching only one cross makes it easy to measure the angle in the wrong direction.
5. Step 5: Choose the ratio — SOH CAH TOAAfter labelling the triangle, decide how the known and unknown sides relate to the angle: use sine when the opposite side and the hypotenuse are involved, cosine for adjacent and hypotenuse, tangent for opposite and adjacent — the chant SOH CAH TOA. Choosing correctly matters because each ratio links exactly three quantities, and two knowns give the third. The hypotenuse is always the longest side opposite the right angle, while the opposite and adjacent sides change with the angle in use, so confirm the right angle before naming the sides.
6. Step 6: Solve, check, and state the answerSolve the equation with your calculator, confirming it is in DEGREE mode before you start. Every answer needs a unit — metres for lengths, degrees for angles — rounded as the question demands, such as three significant figures or one decimal place. Keep extra decimal places during the working and round only at the end, because early rounding shifts the final answer. Finish with a sanity check: a tree should not be taller than a mountain, and the hypotenuse can never be shorter than a leg — if it looks wrong, re-check at once.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Application problems must draw diagram
• Elevation look up, depression look down
• Bearing clockwise from north, three digits
• Find right triangle first can use trig ratio
• Answer remember units