The Law of Cosines is a mathematical formula that connects the sides of any triangle to its angles. Think of it as a super-powered version of the Pythagorean theorem that works on all triangles, not just right triangles.
That’s where a Law of Cosines calculator comes in. It’s a tool that helps you quickly figure out missing side lengths or angles when you’re working with triangles.
You can use the Law of Cosines when you know two sides of a triangle and the angle between them (that’s called “SAS”), or when you know the length of all three sides (“SSS”). This calculator is all you need.
Understanding the Law of Cosines Formula
The Law of Cosines describes the relationship between the sides and angles of a triangle. It’s especially handy when you can’t use the basic trig functions (SOH CAH TOA) because you don’t have a right triangle.
The Standard Formulas
Here are the three standard forms of the Law of Cosines that you’ll use to find the lengths of the sides:
- a² = b² + c² – 2bc cos(A)
- b² = a² + c² – 2ac cos(B)
- c² = a² + b² – 2ab cos(C)
In these formulas:
- a, b, and c are the lengths of the sides of the triangle, and
- A, B, and C are the angles opposite the sides a, b, and c, respectively.
Formulas for Finding Angles
If you’re trying to find an angle, you’ll use these formulas:
- cos(A) = (b² + c² – a²) / 2bc
- cos(B) = (a² + c² – b²) / 2ac
- cos(C) = (a² + b² – c²) / 2ab
Keep in mind that to actually find the angle, you’ll need to take the inverse cosine (arccos or cos⁻¹) of the result. That will give you the measure of the angle in degrees.
Using the Law of Cosines to solve triangles
The Law of Cosines comes in handy when you need to “solve” a triangle, meaning you need to find the lengths of all the sides and the measurements of all the angles. Here are a couple of common situations where you can put the Law of Cosines to work.
Side-Angle-Side (SAS)
If you know the length of two sides and the measure of the angle between them, you can use the Law of Cosines to find the length of the third side. Let’s say you know the lengths of sides a and b, and you know the measure of angle C. You can find the length of side c using this formula:
c² = a² + b² – 2ab cos(C)
For instance, if a = 5, b = 6, and γ (angle C) = 30°, then c would be approximately 3.
Side-Side-Side (SSS)
If you know the lengths of all three sides (a, b, and c), you can use the Law of Cosines to find the measure of any of the angles. To find angle A, you’d use this formula:
cos(A) = (b² + c² – a²) / 2bc
For example, if a = 4, b = 5, and c = 6, then α (angle A) would be approximately 41.41°, β (angle B) would be approximately 55.77°, and γ (angle C) would be approximately 82.82°.
The Ambiguous Case (SSA)
You can use the Law of Cosines when you know two sides and an angle that isn’t between them (Side-Side-Angle), but it’s trickier. This situation can be ambiguous, meaning there might be multiple possible triangles that fit the information, or there might be no triangle at all.
How are the Law of Cosines and the Pythagorean Theorem related?
The Law of Cosines is basically the Pythagorean Theorem, but more generally stated. Remember the Pythagorean Theorem? It says that in a right triangle, a2 + b2 = c2. The Law of Cosines applies to any triangle, not just right triangles.
The Law of Cosines states: c2 = a2 + b2 – 2ab cos(C). Notice that if angle C is 90 degrees (making it a right triangle), then cos(C) = 0. That means the last part of the equation, “- 2ab cos(C)”, becomes zero, and the equation simplifies to c2 = a2 + b2. That’s the Pythagorean Theorem!
So, the Pythagorean Theorem is really just a special case of the Law of Cosines, only applicable to right triangles. The Law of Cosines works for all triangles.
How to use a Law of Cosines calculator
Law of Cosines calculators can help you solve for unknown sides and angles in a triangle. To use one effectively, you’ll need to know some basics.
First, you need to provide three known values. This could be two sides and an angle, or all three sides. The calculator then uses the Law of Cosines formula to determine the missing values.
Most calculators show you the unknown side(s) and/or angle(s) of the triangle. Some may even provide a step-by-step solution, which can be helpful for understanding the process.
To ensure accuracy:
- Clearly state the problem.
- Carefully enter the known values into the calculator.
- Make sure your side and angle labels match the diagram.
Key Takeaways
The Law of Cosines is a powerful tool for figuring out the sides and angles of any triangle, not just right triangles.
When you need to calculate the sides and angles of a triangle, a Law of Cosines calculator can make the process quicker and easier.