Concept 9.2
Choosing the Right Ratio
Source: NCERT Mathematics, Class 1010 minute lesson
Formal statement
In height-and-distance problems, choose the ratio that links the side you know with the side you want: tan when both are legs (height and horizontal distance), sin or cos when the hypotenuse is involved.
The intuition
A tower is seen from 30 m away at an elevation of 60°. You know the horizontal distance and want the height. Both are legs of the right triangle.
The ratio that links two legs is tan: tan 60° = h/30, so h = 30√3 m. sin would drag in the hypotenuse, which nobody asked about.
The attention
Look at which two sides the problem connects, then pick the ratio.
Know distance 30 m, want height h: tan 60° = h/30 not sin 60° = h/30
Look hereNot here
- Look here
- A ratio of the two legs. Height and distance are the two legs, and tan is opposite over adjacent. One equation, one unknown.
- Not here
- A ratio involving the hypotenuse. sin 60° = h/30 treats 30 as the hypotenuse. It isn't: the hypotenuse is the line of sight.
- What it teaches
- Name what you know and what you want before choosing a tool.