Prove that c^2=a^2+b^2-2 a b(cos∠C) is true for any acute ABC. (This formula is called the Law o

step 1 Alright, so for this problem here, we're given a cute triangle ABC. Now notice that the triangle

Prove that c^2=a^2+b^2-2 a b(cos∠C) is true for any acute...

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a specific case of the Law of Cosines that applies when one of the angles of a triangle is exactly 90 degrees. It states that the sum of the squares of the lengths of the two legs equals the square of the hypotenuse. Recognizing this connection helps in understanding how the Law of Cosines reduces to this simpler form under right-angle conditions.

Acute Triangle

An acute triangle is a triangle in which all three interior angles are less than 90 degrees. Understanding the characteristics of an acute triangle is essential for applying the Law of Cosines accurately because the behavior of the cosine function in the range of acute angles (where the cosine values are positive) influences the resultant computation of side lengths.

Cosine Function

The cosine function is a trigonometric function that relates the ratio of the adjacent side to the hypotenuse in a right triangle and extends to define relationships in any triangle. In the context of the Law of Cosines, the cosine value adjusts the relationship between the sides according to the measure of the included angle, effectively compensating for the deviation of the triangle from a right angle.

Law of Cosines

The Law of Cosines is a fundamental theorem in trigonometry and geometry which establishes a relationship among the three sides of a triangle and one of its angles. It generalizes the Pythagorean theorem and works for all types of triangles by incorporating the cosine of an angle, thereby allowing computation of an unknown side or angle when the other components are known.

Transcript

00:01 Alright, so for this problem here, we're given a cute triangle abc.

00:07 Now notice that the triangle is not necessarily isosceles and it's not necessarily equilateral.

00:14 We're just going to use a very plain scalene triangle.

00:19 So given triangle abc, we are going to prove the log cosines saying that c squared equals a squared plus b squared minus 2ab times the cosine of c.

00:34 Now something to notice here is that this little c squared is the side opposite angle c.

00:44 This little a squared, this little a, is the side opposite angle a.

00:50 And last but not least, this little b is the side opposite angle b.

00:55 Just some notation there.

00:57 So let's get started.

00:58 We're looking here at our good old friend cosine and we know that cosine deals with trig.

01:06 And trig generally deals with right triangles.

01:09 So i'm going to draw an altitude here to give us a right triangle to help us out with this.

01:17 And my altitude is the same as my height, so i'm going to name this segment segment h.

01:24 Now segment h has divided segment b into two parts.

01:28 Now again, as i said before, there's nothing here saying that this triangle is isosceles or equilateral, so we don't know anything about the relationship between these two parts here.

01:39 So i'm just going to call this part b1 and this part here b2.

01:45 Again, there is nothing saying that b1 and b2 are equal to each other or one is necessarily greater than the other.

01:53 But what i have created here are two right triangles.

01:57 So i have this first triangle here and i have another one next to it.

02:20 And this is going to give us a few more relationships that we can work with.

02:26 Cool.

02:26 So i'm going to write out a few relationships that are going to be helpful to us.

02:33 First off, the entire length of side b is going to be equal to the length of b1 plus the length of b2.

02:42 It's really important to notice here that this is b sub 1 and b sub 2.

02:46 So b squared does not equal b1 squared plus b2 squared.

02:52 So be really careful about that.

02:55 And with these two triangles, my right triangles, i can do some pythagorean theorem to help come to some conclusions.

03:03 So first off, let's look at our blue one.

03:06 I know that b1 squared plus h squared is going to equal a squared.

03:14 And on my purple triangle on the right, i know that b2 squared plus h squared is going to equal c squared.

03:26 Both of these problems here have my h squared in common.

03:32 So i'm going to rearrange them to set them equal to h squared.

03:37 So h squared is going to equal c squared minus b2 squared.

03:44 And on this side, i'm going to say h squared is equal to a squared minus b sub 1 squared...

Please give Ace some feedback