About Law of Cosines
The law of cosines is a relationship between the lengths of a triangle's sides and the cosine of one of its angles. We may now obtain distance and angle values that would previously be impossible to measure using trigonometry. When computing the third side of a triangle given two sides and their enclosed angle, and when computing the angles of a triangle if all three sides are known, the law of cosines comes into play.
A triangle has six elements: three sides and three angles. To explore the inter-relationship of these parts using the cosine function, we must first grasp the law of cosines formula and its derivation.
What do you mean by Law of Cosines?
The law of cosine aids in determining the link between the lengths of a triangle's sides and the angles' cosine. The Pythagoras theorem, which applies to a right triangle, is generalised by the cosine law in trigonometry.
Law of Cosines:
Statement: The square of any one side of a triangle is equal to the difference between the sum of squares of the other two sides and double the product of the other sides and the cosine angle included between them, according to the law of cosine.
Let a, b, and c be the lengths of a triangle's three sides, and A, B, and C be the triangle's three angles. The law of cosine therefore states: a2 = b2 + c2- 2bc•cosA. As previously established, the Pythagorean theorem is generalised by the law of cosines in trigonometry. What happens if you substitute 90o for the angle in one of the rules? We're left with the Pythagoras theorem since cos 90o = 0.
The cosine rule is another name for the law of cosine. This law can be used to identify missing data in any triangle. For example, if you know the lengths of two triangle sides and the angle between them, this rule can help you find the triangle's third side. In the following sections, we'll look at alternative cosine law formulas and how to discover these missing parameters.
Law of Cosines Formula
There are three laws of cosines, and based on the data, we choose one of them to solve our difficulties.
a2 = b2+ c2 - 2bc•cosA
b2 = c2+ a2 - 2ca•cosB
c2 = a2 + b2 - 2ab•cosC
Finding Missing Length and Angles Using the Law of Cosines
Given the requisite known elements, the law of cosines can be used to compute the missing triangle parameters. Take a look at the steps below to see how to use the cosine law to find a triangle's missing side or angle.
Step 1: Write down the triangle's given data (side lengths and angle measures) and identify the element to be calculated.
Step 2: Use the cosine rule formulas to solve problems.
a2 = b2 + c2- 2bc•cosA
b2 = c2 + a2 - 2ca•cosB
c2= a2 + b2- 2ab•cosC
where the vertices of a triangle are A, B, and C, and their opposite sides are a, b, and c, respectively.
Step 3: Use appropriate units to express the result.
Download free pdf of Law of Cosines Its Use And Solved Examples
