๐ What is the Law of Cosines?
The Law of Cosines is a formula that relates the sides and angles of any triangle. It's particularly useful when you can't use the basic trigonometric ratios (SOH CAH TOA) because you don't have a right triangle. Think of it as a more general form of the Pythagorean theorem.
๐ A Little History
While the modern formulation using trigonometric functions emerged later, the underlying principles of the Law of Cosines have roots in ancient Greek geometry. Euclid's Elements hinted at these concepts, and later mathematicians refined them into the formula we use today.
๐ The Formula and Key Principles
The Law of Cosines can be expressed in three ways, depending on which angle you're trying to find:
- ๐ Finding side $a$: $a^2 = b^2 + c^2 - 2bc \cdot cos(A)$
- ๐ Finding side $b$: $b^2 = a^2 + c^2 - 2ac \cdot cos(B)$
- ๐งฎ Finding side $c$: $c^2 = a^2 + b^2 - 2ab \cdot cos(C)$
Key Principles:
- ๐ It relates all three sides and one angle of a triangle.
- ๐ก It's crucial when you have Side-Angle-Side (SAS) or Side-Side-Side (SSS) information.
- โ๏ธ The angle used in the formula must be opposite the side you're solving for.
๐ Real-World Examples
Let's tackle some word problems. These examples show how the Law of Cosines is used to solve practical problems.
๐ข Example 1: Navigation
A ship travels 60 miles east, then changes direction to $15ยฐ$ north of east and travels another 80 miles. How far is the ship from its starting point?
- ๐บ๏ธ Visualize: Draw a triangle. The sides are 60 miles and 80 miles. The angle between them is $180ยฐ - 15ยฐ = 165ยฐ$.
- โ๏ธ Apply the Law of Cosines: Let $c$ be the distance from the starting point. Then $c^2 = 60^2 + 80^2 - 2(60)(80) \cdot cos(165ยฐ)$.
- โ Calculate: $c^2 = 3600 + 6400 - 9600 \cdot (-0.9659) \approx 19272.64$.
- โ
Solve: $c = \sqrt{19272.64} \approx 138.83$ miles.
๐ฒ Example 2: Surveying
A surveyor wants to find the distance across a lake. She measures the distance from a point to each end of the lake as 250 meters and 320 meters. The angle between these lines of sight is $76ยฐ$. What is the distance across the lake?
- ๐ Visualize: Draw a triangle with sides 250 m and 320 m, and the included angle of $76ยฐ$.
- ๐ Apply the Law of Cosines: Let $d$ be the distance across the lake. Then $d^2 = 250^2 + 320^2 - 2(250)(320) \cdot cos(76ยฐ)$.
- โ Calculate: $d^2 = 62500 + 102400 - 160000 \cdot (0.2419) \approx 126096$.
- ๐ก Solve: $d = \sqrt{126096} \approx 355.09$ meters.
๐๏ธ Example 3: Hiking
Two hikers leave a trailhead. One hiker walks 5 miles due east. The other hiker walks 3 miles in a direction $60ยฐ$ north of east. How far apart are the hikers?
- ๐ถโโ๏ธ Visualize: Form a triangle. One side is 5 miles, another is 3 miles, and the included angle is $60ยฐ$.
- โ๏ธ Apply the Law of Cosines: Let $x$ be the distance between the hikers. Then $x^2 = 5^2 + 3^2 - 2(5)(3) \cdot cos(60ยฐ)$.
- โ Calculate: $x^2 = 25 + 9 - 30 \cdot (0.5) = 19$.
- โ
Solve: $x = \sqrt{19} \approx 4.36$ miles.
โ๏ธ Practice Quiz
- ๐ฒ A triangular garden has sides of length 15 feet, 18 feet, and 22 feet. Find the largest angle in the garden.
- โพ A baseball diamond is a square with sides of 90 feet. What is the distance from home plate to second base? (Use Law of Cosines, even though Pythagorean Theorem works here too!).
- ๐ช A kite is made with two pieces of fabric. One piece has sides 12 inches and 15 inches with an included angle of 110ยฐ. How long is the longest diagonal of this piece?
- ๐งญ A plane flies 200 miles due north, then turns 30 degrees east and flies another 150 miles. How far is the plane from its starting point?
- ๐ A triangle has sides of length 7, 9, and 12. Find the measure of the angle opposite the side of length 12.
- ๐ Two ships leave port at the same time. One travels 10 miles at a bearing of N30ยฐE, and the other travels 8 miles at a bearing of S70ยฐE. How far apart are the ships?
- โฐ๏ธ A surveyor is measuring the distance across a canyon. She stands at one side of the canyon and sights a point on the other side. She then walks 200 feet along her side of the canyon and sights the same point again. The angle between her path and the first line of sight is 35ยฐ, and the angle between her path and the second line of sight is 120ยฐ. How wide is the canyon? (Hint: you will need Law of Sines followed by Law of Cosines).
๐ง Conclusion
The Law of Cosines is a powerful tool for solving problems involving triangles that aren't right triangles. By understanding the formula and practicing with real-world examples, you can master this important concept!