Law of Cosines Calculator

The Law of Cosines Calculator is an online tool for solving triangles using the cosine law. Use our free online calculator to determine the unknown side lengths or angles of a triangle.

How to Use the Law of Cosines Calculator?

• Input

  Find out which sides and/or angles of your triangle you already know. You must have at least one side and either two other sides or a side and an angle. For scenarios with two known sides and one angle, enter the given side lengths and the known angle in the input fields. To find angles with three available side lengths, input them in the corresponding fields.

• Calculation

  After entering all the given values, click the "Calculate" button.

• Result

  The calculator will instantly display the missing side lengths or angles using the law of cosines.

What Is the Law of Cosines?

The Law of Cosines, also known as the Cosine Rule or Cosine Formula, is an important geometry formula that relates the sides and angle of any triangle, be it acute, obtuse, or right-angled. It extends the Pythagorean theorem, a particular case for right triangles.

Formulae

For any triangle with sides $$$a$$$, $$$b$$$, and $$$c$$$ and angles $$$A$$$, $$$B$$$, and $$$C$$$ opposite these sides, the Law of Cosines is given by the following formulae:

$$c^2=a^2+b^2-2ab\cos\left(C\right)$$$$b^2=a^2+c^2-2ac\cos\left(B\right)$$$$a^2=b^2+c^2-2bc\cos\left(A\right)$$

Understanding the Formulae

We will consider only the first equation. All others are similar.

In a right triangle, when $$$C=90^0$$$, $$$\cos\left(C\right)=0$$$, and the formula transforms into the Pythagorean theorem:

$$c^2=a^2+b^2$$

Example

Suppose you have a triangle with the following measurements: side $$$a=4$$$ units, side $$$b=6$$$ units, and $$$C=120^0$$$.

Using the Law of Cosines, let's find the length of the side $$$c$$$:

$$c^2=a^2+b^2-2ab\cos\left(C\right)$$$$c^2=4^2+6^2-2\cdot4\cdot6\cdot\cos\left(120^0\right)$$

Since $$$\cos\left(120^0\right)=-\frac{1}{2}$$$,

$$c^2=16+36+24=76$$

Thus, $$$c=\sqrt{76}=2\sqrt{19}$$$ units.

Law of Cosines Proofs

The Law of Cosines is a fundamental equation in geometry that relates the lengths of the sides of a triangle with the cosine of one of its angles. There are various ways to prove the Law of Cosines, but one of the most straightforward methods is through vector geometry. Another approach is by using the Pythagorean theorem for an oblique triangle. Below are two proofs of the Law of Cosines:

1. Proof Using Vector Geometry

   Let's consider a triangle $$$ABC$$$.

   • Treat the sides $$$a$$$, $$$b$$$, and $$$c$$$ as vectors $$$\mathbf{\vec{CB}}$$$, $$$\mathbf{\vec{CA}}$$$, and $$$\mathbf{\vec{AB}}$$$, respectively.
   • The square of the length of the side $$$c$$$ (opposite the angle $$$A$$$) is $$$\mathbf{\left\lvert\vec{AB}\right\rvert}^2=\mathbf{\vec{AB}}\cdot\mathbf{\vec{AB}}$$$.
   • Use the property of dot product: $$$\mathbf{\vec{CB}}\cdot\mathbf{\vec{CA}}=\mathbf{\left\lvert\vec{CB}\right\rvert}\mathbf{\left\lvert\vec{CA}\right\rvert}\cos\left(C\right).$$$
   • Use the formula for the difference of vectors: $$$\mathbf{\vec{AB}}=\mathbf{\vec{CB}}-\mathbf{\vec{CA}}$$$.
   • Finally, we have $$$\mathbf{\left\lvert\vec{AB}\right\rvert}^2=\mathbf{\vec{AB}}\cdot\mathbf{\vec{AB}}=\left(\mathbf{\vec{CB}}-\mathbf{\vec{CA}}\right)\cdot\left(\mathbf{\vec{CB}}-\mathbf{\vec{CA}}\right)=\mathbf{\vec{CB}}\cdot\mathbf{\vec{CB}}-\mathbf{\vec{CB}}\cdot\mathbf{\vec{CA}}-\mathbf{\vec{CA}}\cdot\mathbf{\vec{CB}}+\mathbf{\vec{CA}}\cdot\mathbf{\vec{CA}}=\mathbf{\left\lvert\vec{CB}\right\rvert}^2-2\mathbf{\vec{CB}}\cdot\mathbf{\vec{CA}}+\mathbf{\left\lvert\vec{CA}\right\rvert}^2=\mathbf{\left\lvert\vec{CB}\right\rvert}^2-2\mathbf{\left\lvert\vec{CB}\right\rvert}\mathbf{\left\lvert\vec{CA}\right\rvert}\cos\left(C\right)+\mathbf{\left\lvert\vec{CA}\right\rvert}^2$$$.

   This proves the Law of Cosines: $$$c^2=a^2+b^2-2ab\cos\left(C\right)$$$.

2. Proof Using the Pythagorean Theorem

   • Drop a perpendicular from the vertex $$$B$$$ to the side $$$b$$$, dividing $$$b$$$ into two segments. Let $$$D$$$ be the point where the perpendicular meets $$$b$$$. Suppose that the length of the perpendicular is $$$h$$$. Also, let $$$\left\lvert DC\right\rvert=x$$$, then $$$\left\lvert AD\right\rvert=b-x$$$.
   • Now, we have two right triangles: $$$ABD$$$ and $$$CBD$$$.

   • Applying the Pythagorean theorem to the triangle $$$CBD$$$ gives the following equation:

     $$h^2=a^2-x^2$$

   • Similarly, from the triangle $$$ABD$$$ we have

     $$h^2=c^2-(b-x)^2$$

   • Equating the two values of $$$h^2$$$ gives the following:

     $$a^2-x^2=c^2-(b-x)^2$$$$a^2-x^2=c^2-b^2+2bx-x^2$$$$c^2=a^2+b^2-2bx$$
   • From the triangle $$$CBD$$$, according to the definition of cosine, $$$\cos\left(C\right)=\frac{x}{a}$$$ or $$$x=a\cos\left(C\right)$$$.

   • Plugging in our value for $$$x$$$ will give that

     $$c^2=a^2+b^2-2ab\cos\left(C\right)$$

   This completes the proof using the Pythagorean theorem.

These proofs, among other things, emphasize the importance of the Law of Cosines and its fundamental role in geometry.

Why Choose Our Law of Cosines Calculator?

• High Accuracy

  Precision matters in mathematical calculations. Our calculator provides accurate results, taking the guesswork out of solving triangle problems and reducing the chance of errors.

• Fast Calculations

  Time is invaluable. Our tool provides instant results, making it perfect for assignments or just quick checks.

• User-Friendly Interface

  Our calculator is designed with users in mind. With its intuitive layout and straightforward instructions, anyone, from students to professionals, can navigate and use the tool with ease.

• Versatility

  Whether you're dealing with an acute, obtuse, or right triangle, our calculator handles all of them. It can determine side lengths or angles based on the data you provide.

• Educational Value

  Beyond just providing answers, our calculator helps in understanding the Law of Cosines better. The breakdown of the calculation process makes it an excellent tool for learners.

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