Friday
Friday: Triangles Everywhere
Comet drops onto the paint-splattered couch. "Right triangles are everywhere now. The slide at the park. A kite string."
"The shadow of the pole from week 6," Wren adds. "Sun angle, shadow, pole height. Same three sides."
"A wheelchair ramp at the library door," Comet says. "Low angle, long run. The tangent has to be small."
Nova projects a kite on a string with a dotted triangle under it. "Would you like a hint? Which side is the string?"
"The hypotenuse," Comet says. "And the angle is at my hand, so the height is string times sine."
"Review today," Wren says. "Our designer, three ratios, four moves, and Pythagoras as the check."
"Then next week we chalk the turntable," Comet says. "Circles. Finally."
Right triangles in a day
| Where | Known | Ratio that helps |
|---|---|---|
| a kite on a string | string length and the angle at your hand | sine gives the height |
| a ramp at a door | rise and run | tangent gives the angle |
| a shadow on the ground | shadow length and the Sun's angle | tangent gives the height |
| a ladder against a wall | ladder length and foot distance | cosine gives the angle |
| a hill path | path length and height gained | sine gives the slope angle |
Every row is one right triangle. Name the sides from the angle you know, and the ratio picks itself.
The week in five lines
- In a right triangle the side ratios depend only on the acute angle, because such triangles are similar.
- sin = opposite ÷ hypotenuse, cos = adjacent ÷ hypotenuse, tan = opposite ÷ adjacent.
- The sine of an angle equals the cosine of its complement, because the two acute angles add to 90°.
- One side and one angle: pick the ratio with both sides, write the equation, solve.
- Two sides: the Pythagorean theorem gives the third, and a ratio gives the angle.
- Read the question.
- Tap your answer.
A kite string 20 meters long makes a 35° angle with the ground. How high is the kite, in meters, to 1 decimal place?
A pole's shadow is 2.5 meters long when the Sun is 60° up. How tall is the pole, in meters, to 1 decimal place?- ?Solve for the unknown side
- ?Mark the angle you know and label the sides opposite, adjacent and hypotenuse
- ?Pick the ratio that uses both sides and write the equation
- ?Circle the side you know and the side you want
- ?Check that the hypotenuse is the longest side, or check with the Pythagorean theorem
| Week review | True or false? |
|---|---|
| The cosine of an angle is adjacent over hypotenuse. | ? |
| A bigger right triangle with the same angle has a bigger tangent. | ? |
| sin 40° equals cos 50°. | ? |
| The tangent of an angle can be bigger than 1. | ? |
| The hypotenuse can be shorter than a leg. | ? |
| 1 = 1 | ? |
Measure two of its sides and find the angle. Then check it with your angle finder.
Great review, designer. Tomorrow is Shop Day: you run an angle finder survey for your family.