Cosine Rule

Cosine Rule for Angles

The cosine rule for non right-angled triangles finds a missing side, or an angle. It works for any triangle and will find the missing sides and angles. This rule also works for obtuse and isosceles triangles. You can use this when you have three sides and no angles, or just an angle and two sides. In its rearranged form, it can also find angles given three sides. It is a useful formula when you need to find the area of a triangle and need to calculate the missing side. In physics, it helps to calculate vectors and bearings.

Formula / Equation

Formula

As you can see, side a is opposite angle A, therefore side b, and side c are adjacent to angle A. To find the angle you must have all three sides. To find a side you must have one angle and two sides.

Rearranged

Rearranged formula for angle.

To find an angle, this is the best way to rearrange the formula for cos A. You do not need to memorise this as it is a simple equation to transpose and can be done in your head within seconds. When you have three sides and need to find the angle, this formula is useful. A missing angle usually involves a little transposition, or you could just refer to this page when required.

How to find angle: As you can see from the formula, when you have 3 sides, finding the angle is simple and you just plug in the values of the sides, and use the inverse cos function to give you the angle in degrees.

Rearranged for Sides

Rearranged for side.

To find a side, this is the best way to rearrange the formula. This formula finds the unknown side. The side it finds is opposite to the known angle A.

Questions

Here are some simple GCSE style questions and answers that you can use as a part of your online worksheet. These are very simple practice questions to get an idea of how to use the rule.

Question 1

Question 1

Your friend Alan has decided to build a large triangular shaped swimming pool. He has designed it to be as large as possible to cover the unused land he currently has on his property.

He plans to sit at one corner of the pool, drink a couple of colas, kick back, and relax after a hard days wheeling and dealing.

Being a stickler for money and precision, he wants to know the precise angle-of-view he can expect to have from this design.

Answer

When you have three sides, it is possible to find the angle. Let us assume that the unknown angle A is x. We already have the three sides of the triangle, where a = 50, b = 66, and c = 55. Therefore it is a simple matter of plugging the values into the equation to find x.

Angle x is therefore 47.75°.

Question 2

Question 2

You are designing the acoustic system of a large hall. You need to find the coverage that one speaker will have and whether everyone in the back row seats will be able to hear the sound.

The speaker’s specification states that it can radiate sound at an angle of 60° maximum. On full power, the sound can reach a maximum of 100 metres in any direction where its acoustic pressure will be one third of the maximum. You need to know how far the sound will spread along the back seats.

Answer

The sound waves radiate at an angle of 60° and travel 100 m on both sides of the triangle. This problem poses an isosceles triangle because two of the sides are the same length.

We know that angle A = 60°, a = x, b = 100 m, and c = 100 m. Using the rearranged formula will find the unknown side x. You just plug in the values into the formula, and the answer is 100 metres.

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