The big picture
Here is the track, one idea to a line.
- The angle you lift: elevation opens up from the horizontal. Height = walk × tan.
- The angle you lower: depression opens down from the horizontal. On level ground, the object reads the same angle as elevation.
- Which ratio: name the side you know and the side you want. That pair picks sin, cos or tan.
- Two posts, one tower: two readings give two heights. The tower has one height, so they must match.
- One vantage, two lines: statue = whole column − pedestal.
- Up and down: one window adds the pieces above and below.
- Finding the angle: tan of the angle = height ÷ shadow. Match the answer to a known tan value.
- The whole solve: read, draw, name, pick, divide, check.
| Idea | Remember | Example |
|---|---|---|
| Elevation | Height = walk × tan of the angle of elevation. | 30 m walk at 60° gives 30√3 m |
| Depression | On level ground, depression from the top = elevation from the object. | 45° down means 45° up at the boat |
| Which ratio | Name the side you know and the side you want. That pair picks sin, cos or tan. | Height and walk: tan |
| Two posts | Two readings, one tower: the two heights must match. | 60 m gap: the near post is 30 m out |
| One vantage | Statue = whole column − pedestal. | 30√3 − 10√3 = 20√3 m |
| One window | Tower = piece above + piece below. | 12√3 + 36 m |
| Finding the angle | tan of the angle = height ÷ shadow. | 10 ÷ 10√3 = 1/√3, so 30° |
| The order | Read, draw, name, pick, divide, check. | Tank 30 m away at 60°: tan gives 30√3 m, and 3600 = 60² checks it |
Chapter 9 is Some Applications of Trigonometry. Open it and read every problem the way this track taught you. Draw the horizontal first. Name the sides. Pick the ratio.
Keep one line: 30 m at 60° gives 30√3 m of tower.