Hello math bugs(🐞) & hivers(🐝)
I hope you are strong and stout and doing good in life.
Well come back to another decent problem. We need to find out the angle B. What's provided is just three sides of a triangle. We know angles of a ∆ and it's sides has a relation. I won't do it only using geometry. I will take help both of geometry and trigonometry. To do that we need to know cosine rule. And before finding the angle , we gonna prove cosine rule first.
Let's find cosB first:
You can see a triangle below whose one of the heights is h which touches BC and thus it divides ∆ ABC into two part. One is in pink colour and the other one is Balck. If we use Pythagorean triplet in both of them, we find h² in each cases. If we compare the two equation we can find value of x in terms of a,b and c. Let's check it in the following figure:
After finding the value of x we can put the value in cosB equation got from pink triangle. Check it in the following figure.Further calculating we can say that cosB equals to ( a² - b² + c² )/2ac.
The same way we can also find cosA and cosC also. If we wanna cram it, we have to keep it in mind when we find cosB,in the NEUMERATOR a² and c² will come in plus sign and b², opposite side taken angle B comes in minus sign and in the DENOMINATOR just comes twice of product of a & c ( no b).
.So, thus cosA equals to (-a² + b² + c²)/2bc and cosC equals to (a² + b² - c²)/2ab. Easy to cram.
Now it's time for finding the solution. In the question angle B is asked. So we gonna use sine rule of cosB. Check details below:
NONTE:
✅ Try to use it while it is a isosceles triangle cause two equal side get cancle.
✅ When you are asked to find out a side of triangle , use cosine rule to have a easy calculation.
✅ In the rule only the side comes in minus is opposite to the angle taken.
If you wanna know about sign rule, the link is here from my previous post.
Thank you so much for stoping by friends.
I hope you have liked my post.
Have a good day.
All is well.
Regards: @meta007