### Video Transcript

π΄π΅πΆ is a triangle where π equals 13 centimetres, π equals 10 centimetres, and cos of πΆ equals 0.2. Find the value of π, giving your answer to three decimal places.

First of all, thereβs an important distinction to be made here between the uses of the letter π. Remember lower case letters represent sides and uppercase letters represent angles. First, letβs sketch this triangle to help visualise the problem a little more clearly.

The triangle looks something like this. So weβve been given the lengths of two the sides and some information relating to the size of the included angle. Weβve then been asked to calculate the length of the third side. All of this information suggest that we need to use the law of cosines for this problem.

Now the law of cosines, remember, is this: π squared is equal to π squared plus π squared minus two ππ cosine of π΄, where the lower case letters represent sides and the uppercase π΄ represents an angle. However, this former isnβt particularly useful here as it isnβt cosine of angle π΄ that weβve been given, itβs cosine of angle πΆ.

So Iβm going to rewrite this but Iβm going to swap π΄s and πΆs around. So everywhere that thereβs a lowercase π, Iβm gonna replace it with a lowercase π. Lowercase πs will be replaced with lowercase πs. And the uppercase π΄ will be replaced with an uppercase πΆ.

This gives me an alternative specification of the law of cosines, but where itβs π squared that weβre calculating. And it tells me that π squared is equal to π squared plus π squared minus two ππ cosine of πΆ.

Remember sides π, π, and π are the sides opposite the angles with the same letter. So Iβve marked them in orange on the diagram. Now weβre ready to substitute the lengths of sides π and π and the value of cos πΆ in this question.

So we have that π squared is equal to 10 squared plus 13 squared minus two multiplied by 10 multiplied by 13 multiplied by cos of πΆ, which remember is 0.2. Evaluating each of these terms gives π squared is equal to 100 plus 169 minus 52.

This simplifies to π squared is equal to 217. Next, in order to find the value of π, we need to take the square root of both sides of this equation. So π is equal to the square root of 217.

And as a decimal, this is 14.730919. The question has asked for this value to three decimal places. So we need to round it. So we have that the value of π to three decimal places is 14.731 centimetres.