Big triangle, small triangle, same ratio
The ratios depend on the angle only, not on the size of the triangle.
Two ladders
A short ladder and a tall ladder lean against a wall at the same angle with the ground. Each ladder, the wall and the ground make a right triangle.
Divide the height each ladder reaches by the length of that ladder.
Is this ratio the same for the short ladder and the tall ladder?
What this lesson covers
The idea
The trigonometric ratios of an angle remain the same whatever the lengths of the sides of the right triangle, because right triangles with the same acute angle are similar.
Two ladders
A short ladder and a tall ladder lean against a wall at the same angle with the ground. Each ladder, the wall and the ground make a right triangle.
Divide the height each ladder reaches by the length of that ladder.
Is this ratio the same for the short ladder and the tall ladder?
- Yes, it is the same
- No, the tall ladder gives a bigger ratio
- No, the short ladder gives a bigger ratio
Move the points
Points P, C and Q sit on one slanted line from A. Each makes its own right triangle. Drag them, change the angle, and switch between sin, cos and tan.
Only the angle matters
The values of the trigonometric ratios of an angle do not vary with the lengths of the sides of the triangle, if the angle remains the same.
Why? The triangles share angle A and each has a right angle, so they are similar (AA). Corresponding sides are in proportion: AM/AB = MP/BC = AP/AC. That gives MP/AP = BC/AC = sin A, and the same for cos A and tan A.
Notes
The trigonometric ratios of an angle stay the same whatever the size of the right triangle, because right triangles with the same acute angle are similar.
Check yourself
A triangle has opposite side 3 and hypotenuse 5. A bigger right triangle has the same angle A and hypotenuse 10. How long is its opposite side?
Answer: 6
sin A = 3/5 for both triangles. Opposite / 10 = 3/5, so the opposite side is 6.
In a right triangle, sin A = 0.6 and the hypotenuse is 20 cm. How long is the side opposite to A (in cm)?
Answer: 12
opposite = sin A × hypotenuse = 0.6 × 20 = 12 cm.
Triangles APM and ACB share the angle A and both have a right angle. What does that tell us?
In a right triangle the opposite side is 2 and the adjacent side is 4. A bigger right triangle has the same angle A and adjacent side 10. How long is its opposite side?
Answer: 5
tan A = 2/4 = 1/2. So opposite / 10 = 1/2 and the opposite side is 5.
- They are congruent.. They need not be the same size, so they need not be congruent.
- They are similar, so their sides are in proportion. — correct. Yes! Two equal angles make them similar (AA), so AM/AB = MP/BC = AP/AC.
- They have the same hypotenuse.. The hypotenuses AP and AC are different lengths.