Tuesday
Tuesday: Two Sides and the Angle Between
Comet ties two ropes to the dock corner. One reaches the buoy at 8 meters, the other a post at 11 meters.
Wren lays the protractor at the corner. "52 degrees between the ropes. How far is the buoy from the post?"
"Law of Sines," Comet says, then stops. "No. I know two sides and the angle between them. No matching pair."
Nova projects the triangle with a dashed height. "Would you like a hint? Drop the height and use Pythagoras twice."
Wren works it through. "The third side squared is 8² plus 11² minus a correction. Two times 8 times 11 times cos 52°."
"About 8.8 meters," Comet says. "Shorter than 11, longer than the gap if the ropes lay flat."
"Two laws now," Wren says. "The trick is knowing which one fits."
The Law of Cosines
Take any triangle with sides a, b, c, where C is the angle between a and b. Then c² = a² + b² - 2ab cos C.
When C = 90°, cos C = 0 and the correction vanishes. The law becomes c² = a² + b², the Pythagorean theorem.
When C is less than 90°, the correction shrinks c. When C is more than 90°, cos C is negative and c grows.
Check with a right triangle: a = 6, b = 8, C = 90° gives c = 10. That is the 6-8-10 triangle.
Which law? Compare two ways
Way one, the Law of Sines, needs a side and the angle across from it, plus one more part. ASA and AAS fit.
Way two, the Law of Cosines, needs two sides and the included angle (SAS), or all three sides (SSS).
For SSS, solve the law for the cosine: cos C = (a² + b² - c²)/(2ab). Then arccos gives C, obtuse or not.
| What the crew measured | Name | First law to use | Why |
|---|---|---|---|
| two angles and the side between them | ASA | Law of Sines | find the third angle, then a side-angle pair |
| two angles and a side not between them | AAS | Law of Sines | a side and its own angle are already a pair |
| two sides and the angle between them | SAS | Law of Cosines | no side has its own angle yet |
| three sides | SSS | Law of Cosines | solve for a cosine, then arccos |
A solved problem to study: the three stakes
- Three stakes on the lawn are 7, 9 and 12 meters apart. Find the angle across from the 12 m side.
- Law of Cosines, solved for the cosine: cos C = (7² + 9² - 12²) / (2 × 7 × 9).
- The top is 49 + 81 - 144 = -14. The bottom is 126. So cos C = -0.111.
- A negative cosine means an obtuse angle. arccos gives C ≈ 96.4°.
- Check: the biggest side, 12, faces the biggest angle. 96.4° is more than 60°, so it is the biggest. Good.
- Read the question.
- Tap your answer.
Two ropes from the dock corner run 8 m to the buoy and 11 m to a post, 52° apart. How far is the buoy from the post, to 1 place?
Three stakes are 7, 9 and 12 meters apart. What is the angle across from the 12 m side, to 1 place?
Two sides of a triangle are 5 m and 9 m, and the angle between them is 110°. Find the third side to 1 place.
Three stakes are 7, 9 and 12 meters apart. What is the angle across from the 7 m side, to 1 place?- Read the question.
- Tap your answer.
| Statement | True or false? |
|---|---|
| The Law of Cosines with a 90° angle is the Pythagorean theorem. | ? |
| A negative cosine from the SSS formula means the angle is obtuse. | ? |
| SAS can be solved with the Law of Sines first. | ? |
| 49 + 81 - 144 = -14 | ? |
| Sides of 3, 4 and 9 meters make a triangle. | ? |
Measure the angle across from the 12 centimeter strip. Compare with the crew's computed angle.
Excellent. Tomorrow the tape and protractor come out in the Boathouse Lab, and you sight a target of your own.