Solve each triangle A B C that exists. See Examples 4-6 A=38^∘ 40^', a=9.72 m, b=11.8 m

Name: Problem 45, Chapter 8: Precalculus Student Solutions Manual 5th
Uploaded: 2020-08-04T15:26:29-08:00
Duration: 6 min 34 s
Channel: Jacquelyn Trost
Description: Solve each triangle A B C that exists. See Examples 4-6 A=38^∘ 40^', a=9.72 m, b=11.8 m

step 1 Let's start this problem by sketching our triangle. That's a good way to see what the triangle l

Solve each triangle A B C that exists. See Examples 4-6...

Key Concepts

Existence and Uniqueness of Triangles

This principle involves determining whether a triangle can be constructed from the given data, particularly in SSA situations where the ambiguous case might yield different possibilities. It combines the use of the Law of Sines and geometric constraints to decide if the provided measurements lead to one unique triangle, two possible triangles, or no valid triangle at all.

Triangle Angle Sum Theorem

The Triangle Angle Sum Theorem states that the sum of the internal angles of any triangle is always 180 degrees. This concept is essential when solving a triangle because, once some angles are known, the remaining angle can be computed by subtracting the sum of the known angles from 180 degrees.

Ambiguous Case (SSA Condition)

This concept arises when a triangle is given by two sides and an angle that is not the included angle. In such cases, it is possible that no triangle exists, one unique triangle exists, or two distinct triangles exist. The problem must be carefully analyzed to determine which scenario applies based on the relationship between the given sides and the sine of the given angle.

Law of Sines

The Law of Sines is used to relate the ratios of sides of a triangle to the sines of their opposite angles. It is especially useful in solving triangles when provided with an angle and its opposing side, along with another side (the SSA condition). This law permits the calculation of unknown angles or sides and helps in identifying potential ambiguous cases.

Transcript

00:01 Let's start this problem by sketching our triangle.

00:04 That's a good way to see what the triangle looks like, what we know, and what we need.

00:09 I'm given an angle a of 38 degrees, 40 minutes.

00:18 I know that side a, which will be over here, opposite on the triangle, is 9 .72 meters.

00:27 And side b is 11 .8 meters.

00:32 Well, angle b will be opposite side b.

00:35 So angle b is going to be somewhere here along the base of my triangle, and angle c will be here at the top.

00:42 The first thing i need to figure out is how many triangles do i have? do i have zero, one, or two? to figure that out, we're going to compare side a with the height of this triangle, that minimum distance between angle c and the side a .b.

01:04 To find what that height is, i'm going to use the definition of sign.

01:10 So sign of angle a, which is 38 degrees 40 minutes, equals opposite over hypotenuse.

01:19 And when i evaluate that, that gives me a height for this triangle of 7 .37 meters.

01:27 Okay.

01:28 A is big enough.

01:30 It is bigger than that height and smaller than the other side we know.

01:34 Side b.

01:36 Therefore, i have two possibilities.

01:40 Either it can be to the left or to the right of that height that we drop down.

01:49 So i have two possibilities for b.

01:53 I have b sub 1 or b sub 2.

01:57 That's going to give me two options for b that i will need to find, two corresponding options for c and two corresponding options for side c.