What is the value of each of the angles of a triangle whose sides are 95,150, and 190 cm in leng

What is the value of each of the angles of a triangle whose...

Key Concepts

Law of Cosines

The Law of Cosines is a fundamental trigonometric relationship in any triangle which relates the lengths of its sides to the cosine of one of its angles. It is particularly useful in cases where the triangle is not right-angled, allowing one to calculate an angle when all three side lengths are known, or a side when two sides and the included angle are given.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as the arccosine, are used to determine the measure of an angle from the value of its cosine. After applying the Law of Cosines to compute the cosine of an angle, the arccos function helps in converting that cosine value into the corresponding angle measurement.

Triangle Angle Sum Theorem

The Triangle Angle Sum Theorem states that the sum of the interior angles in any triangle is always 180 degrees. This property is often used to verify the calculations of individual angles, ensuring consistency in problems where all angles of a triangle are determined.

Transcript

00:01 Alright, so we are going to have some fun with the law of cosines for this problem.

00:07 So we have a triangle, which has sides of 90 centimeters, 150 centimeters.

00:24 There we go, centimeters, and 190 centimeters.

00:29 All right, and we will also, just for convenience, we will label these sides with letters.

00:34 So c, a, and b.

00:40 And as a means of conventions, if we have lowercase b for this side, we're going to have a capital b for the angle opposite of it.

00:50 So if this side is a, then this side is going to be capital a.

00:56 And this side right here, this angle will be a capital c.

01:00 All right.

01:02 So the lot of cosines is this equation right here, c squared is equal to a squared plus p squared minus 2 a b cosine of the angle c all right and these this lettering right here is somewhat arbitrary um i could have uh for example switch can call this angle c and this angle a or switch a and b right here.

01:39 So that is slightly arbitrary.

01:43 The important thing is that if this is your angle b, then you have to call this one a and c or a and c.

01:50 You can't have two angle bs or two angle a's or, you know, any of that crazy stuff.

01:57 All right.

01:57 With this equation right here, then we can solve for angle c.

02:02 And that is going to be our goal so that we can find the a's.

02:05 One of the angles on this triangle.

02:08 We need to find all of them eventually, but we'll start with one just to start simple, you know.

02:14 Alright, so i'm gonna subtract this term from both sides and add c squared to both sides.

02:20 Hope i can do that whole in one step right here without doing anything too crazy, too confusing.

02:31 A squared plus b squared and minus c squared, because i minus this term from both sides.

02:39 All right, and so for this c right here, the c is the angle and the c is the side.

02:50 So i'm going to divide both sides by 2ab.

02:53 So cosine of the capital c, which is the angle, so you go to a squared plus b squared minus little c because it's the side over 2ab.

03:11 All right now i can take the inverse cosine of both sides all right and plugging that into your calculator you will find that c big c is equal to 27 .6 degrees so if you plug in let's see we called this side of a right here so if you plug in 150 centimeters squared plus 190 centimeters squared 90 centimeters squared over 2 times 150 times 190 you will get 27 .6 degrees for this angle right here...

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