Law of Cosines

Law of Cosines

The law of cosines is a mathematical principle in Trigonometry. It establishes the relationships between the angles' cosine value and the length of the sides. The square on one side of a triangle stands equal to the difference in the sum of the squares on the other two sides. The cosine is also two times the product of its other sides with the cosine angle. If a, b and c are the lengths of the three sides with A, B, and C its three angles, then the law of cosine stands as a² = b² + c² – 2bc.cosA.

Applications on the law of cosines

Now let’s look at the various practical, real-life applications of the law of cosines. The sine and cosine values can help find the value of unknown measurements. These principles can enable you to obtain the value of oblique triangles. The cosine law can help you find the value of a side with the given angle between the opposing sides. You can also use the law of cosines to find the angle of all three sides. Sometimes you may have to work with several triangles using a common angle or edge. The right triangles use Pythagorean Theorems, inverse ratios, and trigonometric ratios.   

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Aircraft heading to counterwind

The aerofoil design follows the aerodynamic principle containing Bernoulli’s principle, the Coanda effect, and Newton's third law of motion. These three unique and individual laws work together to achieve lift.Pilots consider the wind speed and the weather conditions to fly on a suitable flight path. It helps them to take advantage of the most effective route.   

When an aircraft takes off, the headwinds blowing against it increase the lift. Thus, a shorter runway with less ground speed can achieve lift-off for the airplane. Likewise, the plane requires less runway coverage with reduced ground speed on a touchdown during landing.  

Statement and proof of the theorem

Let’s look at the statement and proof of the law of cosines theorem.

According to the law of cosines, c² = a² + b² - 2ab cos (C)

Pythagoras Theorem suggests that a² + b² = c² 

Suppose “abc”: a² + b² = c²

 Another “abc”: 2ab cos (C)

Place them together as a² + b² – 2ab cos(C) = c²

The formula can take shape in any of these forms as given below –

Cos (C) = (a² + b² – c²) / 2ab

Cos (A) = (b² + c² – a²) / 2bc

Cos (B) = (c² + a² – b²) / 2ca

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What is the Law of Cosines?

The law of cosines declares that the square of one side is equal to the squares of the other pair of sides. They’re added together and are subtracted with two times the product value of the two sides with the cosine value of the opposing angle. The law of cosines helps find the value of the remaining three parts when two or more sides have known values.

Therefore, the law of cosines can help find the “angle” of a triangle with the given “length” of the sides.

It’s an unclear case if the angle doesn't lie between the two sides.  

Find missing lengths and angles using the law of cosines

There is a unique characteristic of these trigonometric functions. They can help you find the length of the side of a triangle if you know the proper angle in the triangle. Using the right trigonometric ratio can help you get both - the length and the angle.  

As mentioned earlier, you can use the law of cosines to find unknown lengths and angles since the law of cosines obtains the value of the three remaining parts with the known values of two or more sides.

Let’s look at the formula a² = b² + c² – 2bc.cosA. It uses the two basic formulas, SAS and SSS. SAS stands for side-angle-side, and SSS stands for side-side-side.

The side-angle-side or SAS identifies the length of the two sides and the measure of the included angle.

The side-side-side or SSS indicates that the lengths of all the three sides are given. Due to the lack of a solvable proportion, the Law of Sines becomes ineffective.   

The law of cosine states –

a² = b² + c² - 2bc cos A

b² = a² + c² – 2ac cos B

c² = a² + b² – 2ab cos C

The formula shares a resemblance with the Pythagorean theorem without the third term. Since cosine 90 is 0 when the value of C is 90 degrees, the third term becomes 0.

Are you aware of the effect of the Law of Cosines on the Pythagorean Theorem? The Pythagorean Theorem becomes a unique case for the Law of Cosines. You can state it like

b² = a² + c² – 2ac cos B or

a² = b² + c² – 2bc cos A.

Suppose we consider the value of c as

c² = a² + b² – 2ab cos C

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