/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 30 Use the Law of Sines to solve (i... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 30

Use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places. $$A=58^{\circ}, \quad a=4.5, \quad b=12.8$$

Short Answer

Expert verified
Assuming that the calculated values for Angle B and C both lead to valid triangles, there can be two solutions for triangles with the given sides and angles.

Step by step solution

01

Use the Law of Sines to find Angle B

First, apply the Law of Sines to find Angle B. The Law of Sines states that the ratio of the length of a side to the sine of the angle opposite that side is constant. Therefore, we can write \(\frac{a}{\sin A} = \frac{b}{\sin B}\), which when rearranged becomes \(\sin B = \frac{b * \sin A}{a}\). Substituting the given values for A, a and b, we get \(\sin B = \frac{12.8 * \sin(58^\circ)}{4.5}\)
02

Find the primary value of Angle B

In order to find the value of Angle B we take the inverse sine of the value obtained in Step 1, denoted as \(\arcsin(\sin B)\). However, remember to interpret the solution correctly: sine yields two possible values. Since sine is positive in both 1st and 2nd quadrants, B could be either \(\arcsin(\sin B)\) or \(180^\circ - \arcsin(\sin B)\)
03

Find Angle C for the two potential solutions

Now we know two possible values for Angle B, from there we can find two possible values for Angle C using the fact that the angles of a triangle add up to 180 degrees. Therefore, we can write \(C = 180^\circ - A - B\). Calculate this for both possible values of Angle B to find two potential values for Angle C
04

Check for validity of triangle

It's important to verify whether both solutions form a valid triangle. A triangle is invalid if the sum of two angles is over or equal to 180 degrees or if any calculated angle turns out to be 0 or negative. If either instance applies, discard the corresponding solution and accept only the valid one.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the exact values of the sine, cosine, and tangent of the angle. $$15^{\circ}$$

Determine whether the statement is true or false. Justify your answer. The equation \(2 \sin 4 t-1=0\) has four times the number of solutions in the interval \([0,2 \pi)\) as the equation \(2 \sin t-1=0\).

Solve the multiple-angle equation. $$\sec 4 x=2$$

Find the exact value of the expression. $$\sin \frac{\pi}{12} \cos \frac{\pi}{4}+\cos \frac{\pi}{12} \sin \frac{\pi}{4}$$

(a) use a graphing utility to graph the function and approximate the maximum and minimum points on the graph in the interval \([0,2 \pi),\) and (b) solve the trigonometric equation and demonstrate that its solutions are the \(x\) -coordinates of the maximum and minimum points of \(f .\) (Calculus is required to find the trigonometric equation.) Function $$f(x)=\sin x \cos x$$ Trigonometric Equation $$-\sin ^{2} x+\cos ^{2} x=0$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.