The Cosine rule
Introduction
The Cosine Rule is another fundamental tool used to solve triangles, particularly when the Sine Rule cannot be applied. The Cosine Rule is used in non-right-angled triangles and is especially useful for finding unknown sides or angles when we are given:
1. Two sides and the included angle (SAS).
2. All three sides of a triangle (SSS) when solving for an angle.
The Cosine Rule Formula
To find the side length using Cosine rule
For any triangle \(ABC\) with sides \(a\), \(b\), and \(c\) opposite angles \(A\), \(B\), and \(C\) respectively, the Cosine Rule is expressed as:
\[
a^2 = b^2 + c^2 – 2bc \cdot \cos A
\]
This formula can be rearranged to find , Side \(a\) when given angles \(B\), \(C\), and sides \(b\), \(c\).
The other sides and angles using similar forms:
\[
b^2 = a^2 + c^2 – 2ac \cdot \cos B
\]
\[
c^2 = a^2 + b^2 – 2ab \cdot \cos C
\]
To Find Angles using Cosine rule
The Cosine Rule for finding an angle in a triangle is derived from the standard cosine rule formula. To find an angle, we rearrange the formula as follows:
For a triangle with sides \(a\), \(b\), and \(c\), opposite angles \(A\), \(B\), and \(C\) respectively, the formula to find an angle is:
\[
\cos A = \frac{b^2 + c^2 – a^2}{2bc}
\]
Similarly:
\[
\cos B = \frac{a^2 + c^2 – b^2}{2ac}
\]
\[
\cos C = \frac{a^2 + b^2 – c^2}{2ab}
\]
To find the angle, use the inverse cosine function:
\[
A = \cos^{-1} \left(\frac{b^2 + c^2 – a^2}{2bc}\right)
\]
When to use the sine Rule
– You are given two sides and the included angle (the angle between the two sides).
– Use the Cosine Rule to find the third side.
– You are given all three sides.
– Use the Cosine Rule to find one of the angles.
Interactive Graph
Example 1: Finding Side length
In triangle \(ABC\), Side \(b = 7 \, \text{cm}\), Side \(c = 9 \, \text{cm}\), Angle \(A = 60^\circ\). Find the length of side \(a\).
Example 2: Finding an Angle using the Cosine Rule
In triangle \(DEF\), we know that: Side \(d = 8 \, \text{cm}\), Side \(e = 10 \, \text{cm}\), Side \(f = 12 \, \text{cm}\). Find angle \(D\).
Practice Questions:
Instructions:
- Use the Cosine Rule to solve for the unknown angles or sides.
- For finding angles, use the formula:
\[
\cos A = \frac{b^2 + c^2 – a^2}{2bc}
\] - For finding sides, use the formula:
\[
a^2 = b^2 + c^2 – 2bc \cos A
\]
Remember to use a calculator and round your answers to 2 decimal places where necessary.
Questions:
- In triangle \(ABC\), side lengths \(a = 8\), \(b = 6\), and \(c = 10\). Find angle \(A\).
- In triangle \(XYZ\), side lengths \(x = 12\), \(y = 15\), and \(z = 9\).
- Find angle \(Z\).triangle \(PQR\), the sides \(p = 7\), \(q = 10\), and angle \(R = 65^\circ\). Find the length of side \(r\).
- In triangle \(DEF\), the sides \(d = 14\), \(f = 11\), and angle \(E = 50^\circ\). Find the length of side \(e\).
- In triangle \(LMN\), side lengths \(l = 18\), \(m = 20\), and \(n = 25\). Find angle \(L\).
Here are the solutions for the Cosine Rule worksheet:
1. Given \(a = 8\), \(b = 6\), \(c = 10\), the angle \(A = 53.13^\circ\).
2. Given \(x = 12\), \(y = 15\), \(z = 9\), the angle \(Z = 36.87^\circ\).
3. Given \(p = 7\), \(q = 10\), and angle \(R = 65^\circ\), the length of side \(r = 9.48\).
4. Given \(d = 14\), \(f = 11\), and angle \(E = 50^\circ\), the length of side \(e = 10.91\).
5. Given \(l = 18\), \(m = 20\), \(n = 25\), the angle \(L = 45.49^\circ\).