Cosine Formula – Law or Rule of Cosine Double & Half Angle, Addition

Cosine Formula

In the case of Trigonometry, the law of cosines or the cosine formula related to the length of sides of a triangle to the cosine of one of its angles. To understand the concept better, you can always relate the cosine formula with the Pythagorean theorem and that holds tightly for right triangles. When one measured is 90-degree then writing cosine formula by generalizing Pythagorean theorem actually makes sense.

The cosine formula is also useful for computing the third-side of a triangle where two sides are enclosed together to form a triangle and all three sides of a triangle are known. To solve a triangle, you should first find the length of each of its side and its angles too.

So, we use Sine rule when either two angles or one side is given, if two sides and non-included angle is given. At the same time, cosine rule is used when three sides are given or two sides or one angle is given.

When to use Cosine Rule?

Here are the highlights for your reference —

  • When two sides are given in an included angle i.e. SAS theorem
  • When all three sides are given i.e. SSS Theorem.

Sine of a Half Angle

\[\LARGE \sin \frac{a}{2} = \pm \sqrt{\frac{(1- \cos a)}{2}}\]

Cosine of a Half Angle

\[\LARGE \cos \frac{a}{2} = \pm \sqrt{\frac{(1+ \cos a)}{2}}\]

Tangent of a Half Angle

\[\LARGE \tan \left ( \frac{a}{2} \right ) = \frac{1 – \cos a}{\sin a} = \frac{\sin a}{1 + \cos a}\]

Law of Cosine Formula

\[\large a^2=b^2+c^2-2(bc)+Cos\;A\]
\[\large b^2=a^2+c^2-2(ac)+Cos\;B\]
\[\large c^2=a^2+b^2-2(ab)+Cos\;C\]

Cosine Formula

\[\large a^{2}=b^{2}+c^{2}-2bc.\cos A\]

\[\large b^{2}=a^{2}+c^{2}-2ac.\cos B\]

\[\large c^{2}=a^{2}+b^{2}-2ab.\cos C\]

Based on the Cosine formula, this is true that length of any side of a triangle is equal to the sum of squares of length of other sides minus the twice of their product multiplied by cosine of their inclined angles. Here, the value of cosine rule is true if one of the angles if Obtuse. Students are free to rearrange the Cosine formula to derive further trigonometry formulas from the same.