Monday
Monday: The Ramp Has a Ratio
Comet kneels at the bottom of the access ramp with a tape measure. "The ramp rises 0.9 meters over a run of 3.6 meters."
"So what is its angle?" Wren asks. "The script says the cart has to roll up it without tipping."
Comet holds up a protractor taped to a straw, with a string and a washer hanging from its center. "My angle finder."
She lays the straw along the ramp. The string hangs straight down. "It reads about 14 degrees."
"What do you notice?" Wren asks. "Rise over run is 0.25. Does 0.25 belong to 14 degrees somehow?"
Nova projects two right triangles, one small and one big, with the same tilt. "Would you like a hint? Compare their side ratios."
"Same ratio, any size," Comet says. "Our designer, that ratio has a name. Let us find it."
Three sides, three ratios
Pick one acute angle of a right triangle. The hypotenuse is the longest side, across from the right angle.
The opposite side is across from your angle. The adjacent side is next to your angle, but it is not the hypotenuse.
Three ratios of those sides have names. They are the trigonometric ratios of the angle.
| Ratio | Sides | Short name |
|---|---|---|
| sine of the angle | opposite ÷ hypotenuse | sin |
| cosine of the angle | adjacent ÷ hypotenuse | cos |
| tangent of the angle | opposite ÷ adjacent | tan |
Why the ratio belongs to the angle
Two right triangles with the same acute angle share two angles, so they are similar by AA (week 5).
Similar triangles have proportional sides. So opposite over hypotenuse comes out the same in both, whatever their size.
The crew's two braces show it. Brace A has sides 3, 4, 5 and brace B has sides 6, 8, 10, in decimeters.
| Angle across from the shortest side | Brace A (3, 4, 5) | Brace B (6, 8, 10) |
|---|---|---|
| sine: opposite ÷ hypotenuse | 3 ÷ 5 = 0.6 | 6 ÷ 10 = 0.6 |
| cosine: adjacent ÷ hypotenuse | 4 ÷ 5 = 0.8 | 8 ÷ 10 = 0.8 |
| tangent: opposite ÷ adjacent | 3 ÷ 4 = 0.75 | 6 ÷ 8 = 0.75 |
A solved problem to study
Each row of the table matches. The ratio is a property of the angle, not of the brace. That is what makes trigonometry work.
Wren wants the ramp's angle from its rise and run. Here is his work.
The rise is opposite the angle and the run is adjacent. So the tangent is rise ÷ run: 0.9 ÷ 3.6 = 0.25.
Which angle has a tangent of 0.25? The inverse tangent key on a calculator gives about 14°. The angle finder agreed.
Why does this work? Every right triangle with that angle has the same tangent, so the tangent pins the angle down.
- Read the question.
- Tap your answer.
- Read the question.
- Tap your answer.
| Statement | True or false? |
|---|---|
| The hypotenuse is always the longest side of a right triangle. | ? |
| The sine of an angle changes when the triangle is made twice as big. | ? |
| Two right triangles with the same acute angle are similar. | ? |
| The tangent of an angle is adjacent divided by opposite. | ? |
| 3 ÷ 5 = 6 ÷ 10 | ? |
Measure one acute angle in each with a protractor. Are they the same? Write each triangle's three ratios.
Good start, designer. Tomorrow you get a table of sines, cosines and tangents, and use them to find missing sides and angles.