Tuesday
Tuesday: Solving a Right Triangle
Wren pins a card to the corkboard: a table of sines, cosines and tangents from 10 to 80 degrees.
"Every angle has its three numbers," he says. "Pick the angle, read the ratio, and one side gives you another."
Comet points at the rigging cable. "It runs from the frame top, 5 meters up, down to a floor anchor at 50 degrees."
"What can we make of that?" she asks. "We need the cable length before we order the rope."
Nova projects the triangle. "Would you like a hint? 5 meters is opposite the 50 degree angle. You want the hypotenuse."
"Opposite and hypotenuse, so sine," Comet says. "sin 50° = 5 ÷ cable. Cable = 5 ÷ sin 50°."
"Read the table and check it with the Pythagorean theorem," Wren says. "Our designer, two methods, one answer."
The crew's trig table
These values are the sine, cosine and tangent of each angle, rounded to three places. A calculator gives the same.
| Angle | sin | cos | tan |
|---|---|---|---|
| 10° | 0.174 | 0.985 | 0.176 |
| 20° | 0.342 | 0.94 | 0.364 |
| 30° | 0.5 | 0.866 | 0.577 |
| 40° | 0.643 | 0.766 | 0.839 |
| 45° | 0.707 | 0.707 | 1 |
| 50° | 0.766 | 0.643 | 1.192 |
| 60° | 0.866 | 0.5 | 1.732 |
| 70° | 0.94 | 0.342 | 2.747 |
| 80° | 0.985 | 0.174 | 5.671 |
Read a row: sin 30° = 0.5, cos 30° = 0.866 and tan 30° = 0.577.
Notice sin 30° = 0.5 and cos 60° = 0.5. The two acute angles of a right triangle add to 90°.
The side opposite one angle is adjacent to the other. So the sine of an angle is the cosine of its complement.
Four moves to find a side
- Mark the angle you know. Label the three sides opposite, adjacent and hypotenuse from that angle.
- Circle the side you know and the side you want. Pick the ratio that uses both.
- Write the equation: ratio of the angle = known side over wanted side, or wanted over known.
- Solve for the wanted side, then check that the hypotenuse came out longest.
For the cable: sin 50° = 5 ÷ cable, so cable = 5 ÷ 0.766 = about 6.5 meters.
The anchor's distance from the wall is adjacent: tan 50° = 5 ÷ distance, so distance = about 4.2 meters.
Two methods, one answer
Method 1, trigonometry: the cable is 6.5 meters from the sine, as above.
Method 2, Pythagoras: with legs 5 and 4.2, the hypotenuse is about 6.5 meters. The two methods agree, within rounding.
When you know two sides, Pythagoras is quickest. When you know an angle and one side, use a ratio.
Sometimes you know two sides and want the angle. Form the ratio, then find the angle with that ratio.
In the table, which angle has a tangent near 0.25? None exactly, so the crew used the inverse tangent key: about 14°.
Calculators call these keys sin⁻¹, cos⁻¹ and tan⁻¹, or arcsin, arccos and arctan.
- Read the question.
- Tap your answer.
The cable makes a 50° angle with the floor and reaches 5 meters up the frame. How long is the cable, in meters, to 1 decimal place?
The same cable: 5 meters up at 50°. How far is the anchor from the wall, in meters, to 1 decimal place?
A brace 6 meters long leans at 30° to the floor. How high does its top reach, in meters, to 1 decimal place?- Read the question.
- Tap your answer.
- Read the question.
- Tap your answer.
| Statement | True or false? |
|---|---|
| Sine links the opposite side and the hypotenuse. | ? |
| To find the hypotenuse from the adjacent side and the angle, use tangent. | ? |
| sin 20° equals cos 70°. | ? |
| When you know two sides, the Pythagorean theorem can find the third. | ? |
| 0.5 = 0.5 | ? |
Compare them with 10 × sin 40° = 6.4 and 10 × cos 40° = 7.7 centimeters.
Strong work, designer. Tomorrow is Shop Lab: you build the angle finder and measure something tall without a ladder.