Round answers according to the rounding conventions on page 364. Solve the triangles that exist.

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

Key Concepts

Law of Sines

The Law of Sines is a trigonometric formula that relates the lengths of sides of a triangle to the sines of its opposite angles. It is especially useful in solving triangles when given an angle and its opposite side, along with another side or angle, by forming a proportion that equates the ratio of each side to the sine of its corresponding opposite angle.

Ambiguous Case (SSA Condition)

The ambiguous case arises when two sides and a non-included angle (SSA) are given. Depending on the lengths of the sides and the measure of the provided angle, the data may yield no solution, one unique triangle, or two distinct triangles. It is important to analyze the given information carefully to determine the correct number of possible triangle solutions.

Rounding Conventions

Rounding conventions refer to the standardized rules used to round numerical answers to a specified degree of precision, ensuring consistency and clarity. These conventions dictate how intermediate and final numerical results should be approximated, which is particularly important in trigonometric calculations and the final reporting of side lengths and angle measures.

Transcript

00:01 So given this information about a general triangle, right? any triangle can be labeled with capital letters as angles and sides across from the letters as lowercase.

00:09 So what i'm given here is i'm given an angle and two sides that aren't between the angle.

00:16 So i'm given side side angle.

00:19 Okay, whenever i'm given side side angle, the triangle, i either have zero triangles, one triangle, or two triangles.

00:25 And the first step is, is my angle acuter of twos? well, the given angle is acute.

00:31 So i can have zero, one, or two triangles.

00:33 So that doesn't eliminate any possibilities.

00:36 So first i determine that it's acute.

00:38 Next, i'm going to see the side lanes.

00:40 Is my side that's across from the angle, b, is it less than? what's the relationship between b and c? well, b nine is less than 14.

00:51 So in this case, i'm either going to have zero or two triangles.

00:55 So if i end up figuring out another angle, angle c, then i'll have two triangles, so i have to do 180 minus that triangle.

01:03 Or if i don't find one, then i'll have zero triangle.

01:06 So in this case, i'm just going to solve for angle c.

01:11 If i'm given two side lengths and an angle, i'm going to use law of sign.

01:17 So sign of angle b over side length b equals sign of angle c, which i don't know over 14.

01:26 I want to get sign of c by itself.

01:28 So since i'm dividing by 14, i could cross multiply here, but a trick is i could multiply by 14 to both sides.

01:36 And then to get sign completely by itself, opposite of sine is sine inverse.

01:42 I'm going to take sine inverse of the entire left side.

01:48 Sign inverse of the entire left side.

01:52 So then here i'm going to get making sure my calculator is in degrees.

01:59 I'm going to get about 55 .51 or 56 degrees.

02:03 So this is one of my angles c1.

02:07 To find my other angle c2, i'm just going to subtract 180 from this.

02:12 Defend my other of two single.

02:13 So i'm going to end up having two triangles, triangle one and triangle two...

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