# Law of Cosines Calculator

Law of Cosines Calculator helps to calculate the unknown angle in a triangle when the lengths of all three sides are known. This law helps to relate the sides of a triangle to the cosine of one of its angles.

## What is Law of Cosines Calculator?

Law of Cosines Calculator is an online tool that helps to determine the angle of a triangle by using the law of cosines formula when we know the measure of all three sides. This law can be applied to any type of triangle. To use the * law of cosines calculator*, enter the values in the given input boxes.

### Law of Cosines Calculator

NOTE: Please enter the values up to three digits only.

## How to Use Law of Cosines Calculator?

Please follow the steps below to find the unknown angle using the law of cosines calculator :

**Step 1:**Go to Cuemath's online law of cosines calculator.**Step 2:**Enter the lengths of the sides in the given input boxes.**Step 3:**Click on the**"Calculate"**button to find the unknown angle.**Step 4:**Click on the**"Reset"**button to clear the fields and enter new values.

## How Does Law of Cosines Calculator Work?

The law of cosines says that the square of any one side of a triangle will be equal to the difference of the sum of squares of the other two sides and double the product of these two sides that is further multiplied by the cosine of the included angle. Suppose the three sides of a triangle are given by a, b, and c. Furthermore, let A, B, and C be the angles subtended by these three sides respectively. Then the law of cosines can be given by the following formulas, depending upon which missing value has to be determined.

- a
^{2}= b^{2}+ c^{2 }- 2bc.cosA - b
^{2}= c^{2}+ a^{2}- 2ca.cosB - c
^{2}= a^{2}+ b^{2}- 2ab.cosC

If we need to find the value of A subtended by side a, the steps given below can be followed.

- Square the sides; a
^{2}, b^{2}, and c^{2}. - Take the sum of the other two sides; b
^{2}+ c^{2} - Subtract the value of the square of the side that subtends the unknown angle from this value; b
^{2}+ c^{2}- a^{2} - Divide this value by double the value of the product of the other two sides; (b
^{2}+ c^{2}- a^{2}) / 2bc. - Find the inverse cosine of the value from step 4. The resultant will be the measure of A.

Similar steps can be used to find the values of other angles.

## Solved Examples on Law of Cosines

**Example 1:** The sides of a triangle are given by a = 2, b = 3, c = 4. Find the value of the angle opposite a and verify it using the online law of cosines calculator.

**Solution:**

a^{2} = b^{2} + c^{2 }- 2bc.cosA

A = cos^{-1}[(b^{2} + c^{2} - a^{2}) / 2bc]

A = cos^{-1}[(3^{2} + 4^{2} - 2^{2}) / 2(3)(4)]

A = 28.955 degrees.

**Example 2:** The sides of a triangle are given by a = 3.1, b = 6.3, c = 4.8. Find the value of the angle opposite a and verify it using the online law of cosines calculator.

**Solution:**

a^{2} = b^{2} + c^{2 }- 2bc.cosA

A = cos^{-1}[(b^{2} + c^{2} - a^{2}) / 2bc]

A = cos^{-1}[(6.3^{2} + 4.8^{2} - 3.1^{2}) / 2(6.3)(4.8)]

A = 28.5612 degrees.

Similarly, you can try the law of cosines calculator to find the angle subtended by side a for the following:

- a = 12, b = 5, c = 16
- a = 8.6 , b = 7.4, c = 3.7

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