And the Cosine Rule: Its easier than it seems
The cosine rule has 4 variables or unknowns:
side a, side b, side c and an angle.
\( c^2 = a^2 + b^2 - 2ab\cos C \)
Or, to find an angle:
\( \cos C = \frac{a^2 + b^2 - c^2}{2ab} \)
This means that if you are given 3 of those unknowns, you need to use the cosine rule.
E.g. if you are given 3 sides of a triangle (remember it should not be a right angled triangle.)
Use the cosine rule to find the angle.
Or if you are given 2 sides and the angle IN BETWEEN THOSE 2 SIDES then use the cosine rule to find the 3rd side.
Question 1: Finding a Missing Side
A triangle has sides of length 8 cm and 11 cm, with an included angle of 120°. Calculate the length of the third side to 3 significant figures.
Solution:
We use the cosine rule:
\[ c^2 = a^2 + b^2 - 2ab \cos C \]
Substituting values:
\[ c^2 = 8^2 + 11^2 - 2(8)(11) \cos 120^\circ \]
\[ c^2 = 64 + 121 - 2(8)(11)(-0.5) \]
\[ c^2 = 64 + 121 + 88 = 273 \]
\[ c = \sqrt{273} = 16.5 \text{ cm (to 3 significant figures)} \]
Final Answer: 16.5 cm
Question 2: Finding an Angle
A triangle has sides of length 10 cm, 12 cm, and 15 cm. Find the largest angle in the triangle to 1 decimal place.
Solution:
The largest angle is opposite the longest side, so we find angle A using the cosine rule:
\[ \cos A = \frac{b^2 + c^2 - a^2}{2bc} \]
Substituting values:
\[ \cos A = \frac{10^2 + 12^2 - 15^2}{2(10)(12)} \]
\[ \cos A = \frac{100 + 144 - 225}{240} \]
\[ \cos A = \frac{19}{240} \]
\[ A = \cos^{-1} \left( \frac{19}{240} \right) \]
\[ A = 85.5^\circ \]
Final Answer: 85.5°
Question 3: Word Problem – Bearings
Two boats, A and B, leave a port at the same time.
- Boat A travels 20 km due east.
- Boat B travels 25 km on a bearing of 40° from the port.
Find the distance between the two boats to 1 decimal place.
Solution:
We form a triangle where:
- \( a = 20 \) km
- \( b = 25 \) km
- Included angle \( C = 40^\circ \)
Using the cosine rule:
\[ c^2 = a^2 + b^2 - 2ab \cos C \]
Substituting values:
\[ c^2 = 20^2 + 25^2 - 2(20)(25) \cos 40^\circ \]
\[ c^2 = 400 + 625 - 2(20)(25)(0.766) \]
\[ c^2 = 1025 - 766 = 259 \]
\[ c = \sqrt{259} = 16.1 \text{ km} \]
Final Answer: 16.1 km
Cosine Rule Triangle Calculator - beta mode
Enter two sides and the included angle (in degrees):
More examples and practice questions coming soon!
