Round answers according to the rounding conventions on page 364. Solve the triangles. A=54.32^∘,

Round answers according to the rounding conventions on page...

Key Concepts

Law of Sines

The Law of Sines establishes a proportionality between the lengths of sides of a triangle and the sines of their opposite angles. It is an essential tool for solving triangles when given two angles and one side (AAS or ASA cases) or two sides and an angle not enclosed by them (SSA case). This law allows one to set up ratios that can create equations to solve for unknown angles or sides, enabling a complete solution of the triangle.

Ambiguous Case in SSA

When a triangle is given by two sides and an angle not included between them (the SSA condition), there may exist one, two, or no possible solutions. This situation is known as the ambiguous case. It occurs because the provided data may correspond to different possible configurations of the triangle, and determining the correct solution requires careful application of the Law of Sines and consideration of the triangle’s geometry.

Triangle Angle Sum Theorem

The Triangle Angle Sum Theorem states that the sum of the interior angles of any triangle is always 180 degrees. This fundamental property is frequently used to find a missing angle once the other two are known. Applying this theorem ensures that the computed angles are consistent with the inherent geometry of triangles.

Rounding Conventions

Rounding conventions are systematic rules used to reduce numerical answers to a desired degree of precision, which is important in ensuring that answers are both manageable and understandable. Such conventions, as mentioned in reference materials, detail how to correctly round fractional and decimal values so that final answers accurately reflect the required precision without introducing significant error.

Transcript

00:01 Okay, whenever i'm given information about a triangle, note that any triangle can be written as angles capital a, b, and c, and sides across from the angles, lowercase a, b, and c.

00:10 So i'm given an angle and two sides, not between the angle.

00:15 So i'm given, when i'm given side -side angle information, then i can either have zero triangles, one triangle, or two triangles that are solutions.

00:24 So how do i figure out which one it is? well, first thing i'm going to do is i'm looking at my angle measure.

00:30 Is my angle measure acute or obtuse? well, since it's less than 90 degrees, it's acute.

00:36 Okay? from this, this doesn't tell me much, but since it's acute, i'm going to look at the side across from angle a, lowercase a, and compare it to the other side length.

00:48 I'm giving it.

00:49 Since 24 is greater than 16, i have a is greater than c.

00:53 So what this indicates here is that i will exactly have one triangle.

00:57 If a was less than c, then i could have zero or two triangles.

01:02 Depending on the angle.

01:03 And of course, if the angle was obtuse, then i could either have zero or one triangles.

01:08 So i only have one triangle that's going to satisfy this.

01:10 So what measure i'm going to solve for next? well, i know my angle sideline pair.

01:18 So i'm going to figure out my other angle, sign of c.

01:21 So i'm going to use the law of signs here.

01:24 Sign of c over the sideline c equals sign of the angle a.

01:31 I'll divide it by the side length.