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T4 Cosine rule

 

A non right angled triangle with internal angles A, B and C. The sides opposite these angles are labelled a, b and c respectively. A statement of the cosine rule for side a.

Pythagoras’s Theorem may be used to find the third side in any right angled triangle. The Cosine Rule can be used to solve non-right triangles.

The Cosine Rule

Consider the triangle below:

A non right angled triangle with internal angles A, B and C. The sides opposite these angles are labelled a, b and c respectively.

The angles A, B, C are the angles at the vertices A, B, C respectively. The sides a, b, and c are opposite angles A, B, C respectively.

The Cosine Rule states:

\[\begin{align*} a^{2} & =b^{2}+c^{2}-2bc\cos A\\ b^{2} & =a^{2}+c^{2}-2ac\cos B\\ c^{2} & =a^{2}+b^{2}-2ab\cos C. \end{align*}\]

Note that the side on the left hand side of the equation is opposite the angle listed at the end of the equation:

\[\begin{align*} \boldsymbol{\underline{a^{2}}} & =b^{2}+c^{2}-2bc\,\cos\boldsymbol{\underline{A}}. \end{align*}\]

Use the Cosine Rule when you are given:

  1. two sides and the angle between them, or

  2. all three sides of the triangle.

Examples

  1. Find the value of \(a\) in this triangle

Triangle with internal angle of 83 degrees opposite side a. Length of side on left is 12 and length of side on right is 15.

\[\begin{align*} a^{2} & =b^{2}+c^{2}-2bc\,\cos A\\ a^{2} & =12^{2}+15^{2}-2\times12\times15\times\cos83^{\circ}\\ a^{2} & =144+225-360\times\cos83^{\circ}\\ a^{2} & =369-43.87\\ a^{2} & =325.13\\ a & =18.03 \end{align*}\]

  1. Find the size of angle B in this triangle:

Triangle with internal angles A, B and C. Length of sides opposite these angles are 5,11 and 7 respectively.

\[\begin{align*} b^{2} & =a^{2}+c^{2}-2ac\,\cos B\\ 11^{2} & =5^{2}+7^{2}-2\times5\times7\times\cos B\\ 121 & =25+49-70\times\cos B\\ 121-25-49 & =-70\times\cos B\\ 47 & =-70\times\cos B\\ \frac{47}{-70} & =\cos B\\ B & =\cos^{-1}\left(-\frac{47}{70}\right)\\ B & =132^{\circ}11^{\prime} \end{align*}\]

Exercise

  1. Use the sine OR cosine rule to find the pro-numeral shown:

Triangle with internal angle of 84 degrees. Side a is opposite this angle. Left hand side is of length 23.1. Right hand side length is 19.6.

Triangle with internal angle of 161 degrees. Length of side opposite this angle is 6.6. Other sides are of length b and 2.3.

c)

Triangle with internal angle of 35 degrees. Side opposite this angle is c. Top side length is 13.6. Length of remaining side is 8.2.

Triangle with internal angle of 28 degrees. Opposite side is of length 1.25. Left hand side length is d. Right hand side length is 0.93.

  1. Find the magnitude of the labeled, unknown angle

Triangle with internal angle of alpha degrees. Opposite side is of length 9.2. Top side is of length 15.6. Bottom side is of length 10.1.

Triangle with internal angle beta degrees. Opposite side has length 7.21. Right hand side has length 4.93. Lower side has length 9.99.

1 a) \(28.7\quad\) b) \(4.38\quad\) c) \(8.33\quad\)d) \(1.05\)

2 a) \(\alpha=34.2^{\circ}\quad\) b) \(\beta=42.9^{\circ}.\)

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