Which quotient, which side
Every problem starts with one choice. Which ratio do I use?
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- One triangle: height, walk, line of sight.
- Height and walk: the tangent.
- Height and line of sight: the sine.
- Walk and line of sight: the cosine.
Name two sides first
Pick the ratio that joins your known side and your wanted side.
At the angle of elevation, the height is the opposite side. The walk is the adjacent side. The line of sight is the hypotenuse.
Each ratio joins two sides.
Rule: Name the side you know and the side you want. That pair picks sin, cos or tan.
| Sides named | Ratio |
|---|---|
| Height and walk | tan |
| Height and line of sight | sin |
| Walk and line of sight | cos |
| Two sides, no angle | Pythagoras |
Three examples
A kite string is 60 m long. The string makes 30° with the ground. You know the string and want the height. So use sin. Sin 30° = 1/2. Height = 60 × 1/2 = 30 m.
Now you know the string and want the distance along the ground. So use cos. Cos 30° = √3/2. Distance = 60 × √3/2 = 30√3 m.
A tower is 30 m tall and you stand 30 m away. No angle is given. Use Pythagoras: height² + walk² = line of sight². So line of sight² = 30² + 30² = 1800. Then the line of sight is √1800 = 30√2 m. This works because 1800 = 900 × 2 and √900 = 30.