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# Law of Cosines Calculator

Select the parameter you wish to calculate and enter the required ones in the respective fields. The tool will use the law of cosine to determine the results instantly.

Calculate:

$$A = \cos^{-1} \left[ \dfrac{b^2+c^2-a^2}{2bc} \right]$$

$$B = \cos^{-1} \left[ \dfrac{a^2+c^2-b^2}{2ac} \right]$$

$$C = \cos^{-1} \left[ \dfrac{a^2+b^2-c^2}{2ab} \right]$$

$$a = \sqrt{b^2 + c^2 - 2bc \cos A }$$

$$b = \sqrt{a^2 + c^2 - 2ac \cos B }$$

$$c = \sqrt{a^2 + b^2 - 2ab \cos C }$$

Side a

Side b

Side c

Angle A (α)

Angle B (β)

Angle C (γ)

Table of Content
 1 Law of Cosines Formulas: 2 Law of Cosines for Sides a, b, and c: 3 Law of Cosines for Angles A, B, and C: 4 Describe the Main Triangle Formulas: 5 What is the Role of the Pythagorean Theorem in the Law of Cosines? 6 How we can Prove the Law of Cosines?
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Use a free online law of cosines calculator that helps to find the unknown lengths of sides or angles of a triangle. You can calculate all other remaining sides and measure angles in a triangle by using different forms of cosine law.

## What is the Law of Cosines?

The law of cosines is a collection of formulas that relates the length of sides of a triangle to one of its cosine angles. The cosine law usually preferred when three sides of a triangle are given for finding any angle A, B, or C of the triangle or the two adjacent sides and one angle is given.

The law of cosines formula is a form of Pythagorean Theorem which adapted for use of non-right triangle, but the Pythagorean Theorem only works for right triangles. So, you can use the law of cosine calculator to compute any value of sides and angles of a triangle.

### Law of Cosines Formulas:

If length of sides are a, b, and c opposite to the angles A, B, and C. Then law of cosine express:

$a^2=b^2+c^2−2 \text{ b c } cos(A)$
$b^2=a^2+c^2−2 \text{ a c } cos(B)$
$c^2=a^2+b^2−2 \text{ a b } cos(C)$

### Law of Cosines for Sides a, b, and c:

In order to find any side of a triangle the law of cosines formula transforms if you know two length of sides and the measures of an angle which is opposite to one of them.

$a=\sqrt{b^2+c^2−2 \text{ b c } cos(A)}$
$b=\sqrt{a^2+c^2−2 \text{ a c } cos(B)}$
$c=\sqrt{a^2+b^2−2 \text{ a b } cos(C)}$

Also, if the two lengths of sides and angle is known, then simply add the values into the law of cosines calculator, and let it doe perform calculations.

### Law of Cosines for Angles A, B, and C:

If you know three sides of a triangle then you can use the cosine rule to find the angles of a triangle. So, the solving formula for the angles which are used by the law of cosines formula is:

$A=cos^{−1}[\frac{b^2+c^2−a^2}{2bc}]$
$B=cos^{−1}[\frac{a^2+c^2−b^2}{2ac}]$
$C=cos^{−1}[\frac{a^2+b^2−c^2}{2ab}]$

Example:

In triangle $$∠ ABC, side a = 9 cm, side b = 10 \text{ cm and side} c = 13 cm$$. Find largest angle.

Solution:

The largest angle of a triangle is facing the longest side, C:

$$c^2 = a^2 + b^2– 2ab cos C$$

$$cos C = [\frac{a^2+b^2−c^2}{2ab}]$$

$$cos C = [\frac{9^2+10^2−13^2}{2(9)(10)}]$$

$$= 0.067$$

So the largest angle of triangle ABC is

$$C = 86.2^0$$

However, An online Law Of Sines Calculator helps you to find the unknown angles and lengths of sides of a triangle.

## How Law of Cosines Calculator Works?

The law of cosine calculator helps you to find all unknown missing values of a triangle by using subsequent steps:

### Input:

• First, you need to choose a side or an angle of the triangle that requires from the drop-down menu, then the calculator displays the corresponding form of the law of cosines formula.
• Now, put all the values into a related field.
• Then, select the correlated units.
• After putting all values click the calculate button.

### Output:

The law of cosines calculator displays the following results by using the law of cosine formula:

• The measures of all angles and length of sides.
• As well as, the values of different characteristics of the triangle.
• In the end, it draws a diagram for the given values.

### Describe the Main Triangle Formulas?

The characteristics of Triangle are:

Perimeter P = $$a + b + c$$

Semi-perimeter s = $$0.5 * (a + b + c)$$

Area A = $$\sqrt {s*(s – a)*(s – b)*(s – c)}$$

Radius of circle in the triangle r = $$\sqrt{(s – a)*(s – b)*(s – c) / s}$$

Radius of circle around triangle R = $$\frac{(abc)} { (4AS)}$$

Where:

a = Triangle Side a, b = Triangle Side b, c = Triangle Side c

A = Angle A, B = Angle B, C = Angle C

R = radius of circle around the circle

r = radius of inside circle

P = Perimeter

s = Semi-perimeter

A = Area

### What is the Role of the Pythagorean Theorem in the Law of Cosines?

The law of cosine is a modified version of the Pythagorean Theorem which is used to find unknown values of sides and angles of the non-right triangles.

### How we can Prove the Law of Cosines?

We can prove the law of cosine equation, by using four different following methods:

• The distance formula
• Trigonometry
• Ptolemy’s theorem
• The Pythagorean Theorem

## Final Words:

The law of cosines calculator is 100% free to use for finding the sides and the angles of a triangle. Computing all the trigonometry measurements manually is really complex task, which increases the chance of errors. So by using this handy calculator you can prevent the risk of getting exact values as well as this online calculator assists both students and tutors to solve the law of cosines related problems.

## Reference:

From the source of Wikipedia: Ptolemy’s theorem, Obtuse case, Pythagorean theorem, The distance formula.

From the source of Versity Tutors: Applications of Law of cosines, The Pythagorean Theorem, Two Sides and the Included Angle-SAS.

From the source of Hyper Physics: Law of Cosines, Aircraft Heading to Counter Wind.