The angle owns the ratios
A small ladder and a tall ladder stand at the same angle. Do they share the same sin A?
Loading lab…
Step 1 of 4
- One right triangle at angle A.
- A bigger triangle with the same angle.
- A third size. Same angle again.
- The ratio is the same at every size.
Same angle, same ratio
Yes. Sin A depends on angle A, not on the ladder's size.
Two right triangles with the same angle A are similar. Similar means the same shape, in a different size. Matching sides are in the same position in each triangle. Every side grows by the same number of times.
Rule: Angle A alone decides sin A, cos A and tan A.
Take a 3-4-5 triangle where A faces the side 4. Here sin A = 4/5. Double every side to get 6-8-10. The scale factor is 2, the number each side is multiplied by. Now sin A = 8/10. Divide top and bottom by 2. That is 4/5 again. So the scale factor cancels in the division.
| Size | Opposite | Adjacent | Hypotenuse | sin A |
|---|---|---|---|---|
| ×1 | 4 | 3 | 5 | 4/5 |
| ×2 | 8 | 6 | 10 | 4/5 |