Cosine Formula

The cosine formula, also known as the law of cosines, cosine rule, or al-Kashi's theorem is a trigonometric formula that connects the lengths of a triangle's sides to the cosine of one of its angles. 

The Pythagorean theorem, which only applies to right triangles, is generalised by the cosine formula. 

When two sides of a triangle and their enclosed angle is known, the cosine formula can be used to compute the third side of a triangle, as well as the angles of a triangle when all three sides are known.

The Cosine formula states that the square of the length of any side of a triangle is equal to the sum of the squares of the other sides minus twice the product of the other two sides multiplied by the cosine of angle included between them.

The cosine formula applies to the lengths of a triangle's sides when one of its angles is a cosine angle. We can use this rule to determine the length of a triangle's side of the angle between the sides.

Cosine Formula for Triangle

Consider a triangle ABC whose lengths of the sides are a, b and c as shown in the below figure. The angle between each side is 𝞪, 𝞫 and 𝞬.

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So the cosine formula to find the length of the sides a, b and c is given as follows:

a² = b² + c² - 2 × b × c × cos 𝞪

b² = a² + c² - 2 × a × c × cos 𝞫

c² = a² + b² - 2 × a × b × cos 𝞬

The cosine formula can be written to find the angles 𝞪, 𝞫 and 𝞬 as follows:
\[cos\alpha = \frac{b^{2}+c^{2}-a^{2}}{2\times b\times c}\]

\[cos\beta = \frac{a^{2}+c^{2}-b^{2}}{2\times a\times c}\]

\[cos\gamma = \frac{a^{2}+b^{2}-c^{2}}{2\times a\times b}\]

Proof Of Cosine Formula For Triangle By Using Distance Formula

Consider a triangle with sides of lengths a, b, and c, where 𝞱 is the angle measurement opposite the side of length c. By graphing the components of the triangle's three points, this triangle may be put on the Cartesian coordinate system, aligned with edge a and the origin at C as shown in the figure below.

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The coordinates of points A, B and C are

A = (b cos 𝞱, b sin 𝞱)

B = (a, 0) 

C = (0, 0)

Now applying the distance formula to find the distance between points A and B we get

\[\sqrt{(a-bcos\theta )^{2}+(0-bsin\theta )^{2}}\]

Now squaring on both sides we get

\[c^{2} = {(a-bcos\theta )^{2}+(-bsin\theta )^{2}}\]

Now solving the RHS

\[c^{2} = {a^{2}+b^{2}cos\theta -2abcos\theta +b^{2}sin^{2}\theta }\]

\[c^{2} = {(a-b(cos\theta ^{2}+bsin\theta )^{2}-2abcos\theta }\]

We know that \[cos^{2}\theta +sin^{2}\theta = 1\]

\[c^{2}=a^{2}+b^{2}-2abcos\theta \]

Hence the proof.

Exercise Problems on Cosine Formula

1. In the given figure if a = 4 cm and b = 6 cm and 𝞬 = 30⁰. Find the missing side of the triangle.

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Ans: Here the two sides are given as a = 4 cm and b = 6 cm and the angle between these two sides is 𝞬 = 30⁰.

Now to find the third side we will use the cosine formula as follows:

c² = a² + b² - 2 × a × b × cos 𝞬

On substituting the values we get

c² = 4² + 6² - 2 × 4 × 6 × cos 30⁰

c² = 16 + 36 - 48 × \[\frac{\sqrt{3}}{2}\]

c² = 10.4 cm.

Therefore the length of the missing third side is 10.4 cm.

2. If the lengths of the sides of a triangle is a = 5 cm, b = 8 cm and c = 12 cm. Find the two angles 𝞪 and 𝞫.

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Ans: Here the sides of the triangle are given as a = 5 cm, b = 8 cm and c = 12 cm and we have to find the angles 𝞪 and 𝞫.

We will use cosine formula to find the angle between each side

To find 𝞪

\[cos\alpha = \frac{b^{2}+c^{2}-a^{2}}{2\times b\times c}\]

Substituting the values we get

\[cos\alpha = \frac{8^{2}+12^{2}-5^{2}}{2\times 8\times 12}\]

\[cos\alpha = \frac{64+144-25}{192}\]

\[cos\alpha = \frac{182}{192}\]

Cos =0.953125

\[\alpha = cos^{-1}0.953125\]

\[\alpha = 17.6^{\circ}\]

To find 𝞫 

\[cos\beta = \frac{a^{2}+c^{2}-b^{2}}{2\times a\times c}\]

Substituting the values we get

\[cos\beta = \frac{5^{2}+12^{2}-8^{2}}{2\times 5\times 12}\]

\[cos\beta = \frac{25+144-16}{120}\]

\[cos\alpha = \frac{105}{120}\]

\[cos\alpha = 0.875\]

\[\beta = cos^{-1}0.875\]

\[\beta = 28.96^{\circ}\]

Conclusion

In trigonometry, the cosine function is a periodic function that is particularly essential. The unit circle is the simplest method to comprehend the cosine function. When the lengths of two sides and the measure of the included angle (SAS) or the lengths of the three sides (SSS) are known, the cosine formula for the triangle is used to identify the remaining parts of an oblique (non-right) triangle.