We can use the sine rule and cosine rule to find missing lengths and angles in triangles which aren't right-angled. The difficulty with the sine rule and cosine rule is recognising which one to use. Instead of memorising a formula you need to memorise two patterns. So to start use this simple aid to memory which will make sense as you read on:
- Sine for angle and opposite side
- Cosine for an angle sandwich
If you have the length of a side and its opposite angle then use the sine rule. If you have an angle sandwiched between two sides where you know the length then use the cosine rule. That's it!
This formula is easy to memorise because it is always the side length divided by the sine of the opposite angle. The pattern you're looking for is an angle and its opposite side.
When you see an angle and opposite side use the sine rule
Here is an example to show what's meant by angle and opposite side. We're given an angle of 50 degrees and the opposite side length of 14 cm and another side of 6 cm. We have to work out x the angle opposite 6 cm.
So the approach is:
- Write down known length over the sine of the opposite angle (14/sin 50 in this case)
- Equate it with the other side and the sine of the unknown angle (6/sin x)
- Rearrange to get the sine of the angle as the subject (right hand side is 0.3283)
- Calculate the inverse sine of the right hand side (get 19.17 degrees)
The formula to learn is:\(a^2=b^2+c^2-2bc \cos A\)
It may help to think of this as a corrected form of Pythagoras’ Theorem \(a^2=b^2+c^2\). The “correction” term is the \(-2bc\cos A\). The way to remember this is to visualise the pattern. If we put the equation into words it would be something like this:
- Word Equation: The side I’m working out squared is the sum of the other two sides squared minus twice the product of the other two sides times the cosine of their sandwiched angle.
Notice how the angle looks like it's sandwiched between the two sides that we know. The bread is the two sides we know: 6 cm and 10 cm. The side we're working out is the one that looks like the sandwich filling.
When you see the angle sandwich use the cosine rule
Don't forget to take the square root to work out the length of the side we don't know. In the example above you should get 9.74 cm.
Once you know this pattern you don't have to memorise the other forms of the cosine formula. Just remember which sides are the bread (6 cm and 10 cm in this example) which angle is the filling (70 degrees) and the word equation always works.
There is one variant which you sometimes get asked. If you see all three sides and no angles you can use the cosine rule to find any of the missing angles. Just re-arrange the formula above and you can figure this out for yourself:
As before you don't need to memorise any more equations, just remember that to work out the sandwiched angle you know this pattern:
- Add the squares of the sides adjacent to the angle (the bread, or sides next to the angle).
- Subtract the square of the opposite site (the side opposite to the angle).
- Divide by twice the product of the adjacent sides.
- Take the inverse cosine of the result to get the angle.
If you want to build some triangles and try out your sine rule and cosine rule skills take a look at TrianCal which lets you enter 3 pieces of information (sides and angles) and calculates the rest. And here is a worksheet that has 3 examples of each type of problem that we discussed above with complete solutions so that you can practice.