Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 19: Problem 1881

If \(\Delta \mathrm{ABC}, \mathrm{a}=2, \mathrm{~b}=3\) and \(\sin \mathrm{A}=(1 / 3)\), then \(\mathrm{B}=\) (a) \((\pi / 4)\) (b) \((\pi / 6)\) (c) \((\pi / 2)\) (d) \((\pi / 3)\)

Short Answer

Expert verified
\(B = \frac{\pi}{6}\)

Step by step solution

01

Recall the sine rule

The sine rule for any triangle states that: \[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\] where a, b, and c are the side lengths of the triangle, and A, B, and C are the angles opposite to those sides respectively.
02

Write down the given information

The given information is: a = 2 b = 3 \(\sin A = \frac{1}{3}\)
03

Apply the sine rule to find sine of angle B

Using the sine rule, we have: \[\frac{a}{\sin A} = \frac{b}{\sin B}\] Plug in the given values: \[\frac{2}{\frac{1}{3}} = \frac{3}{\sin B}\] Simplifying the equation, we get, \[\sin B = \frac{9}{2}\]
04

Correct the value of sine of angle B

Since the value of sine cannot exceed 1, the value of sin B above is incorrect.
05

Revisit the sine rule and note the alternative form

The sine rule can also be written as follows: \[\frac{\sin A}{a} = \frac{\sin B}{b}\]
06

Apply the alternative form of the sine rule

Using the alternative form of the sine rule, we have: \[\frac{\sin A}{a} = \frac{\sin B}{b}\] Plug in the given values: \[\frac{\frac{1}{3}}{2} = \frac{\sin B}{3}\]
07

Solve for sine of angle B

Multiply both sides by 3 to solve for sin B: \[\sin B = 3 \cdot \frac{\frac{1}{3}}{2} = \frac{1}{2}\]
08

Find angle B using the inverse sine function

Since \(\sin{B} = \frac{1}{2}\), we find angle B by using the inverse sine function: \[B = \sin^{-1} \left( \frac{1}{2} \right)\] From here, we find that angle B is equal to \(\frac{\pi}{6}\) (option b).

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

If \(\mathrm{K}\left[\sin 18^{\circ}+\cos 36^{\circ}\right]=5\) then \(\mathrm{K}=\) (a) \(2 \sqrt{5}\) (b) \((\sqrt{5} / 2)\) 4 (d) 5

\(0

If \(\sin ^{-1}(1-x)-2 \sin ^{-1} x=(\pi / 2)\) then \(x=\) (a) \(0,(1 / 2)\) (b) \(1,(1 / 2)\) (c) 0 (d) \((1 / 2)\)

If \(\sin \alpha-\sin \beta=m\) and \(\cos \alpha-\cos \beta=n\) then \(\cos (\alpha-\beta)=\) (a) \(\left[\left(2+m^{2}+n^{2}\right) / 2\right]\) (b) \(\left[\left(2-m^{2}-n^{2}\right) / 2\right]\) (c) \(\left[\left(m^{2}+n^{2}\right) / 2\right]\) (d) \(-\left[\left(m^{2}+n^{2}\right) / 2\right]\)

\(\sin \left[\cot ^{-1}\left(\cos \left(\tan ^{-1} x\right)\right)\right]=\) (b) \(\left.\sqrt{[}\left(x^{2}+1\right) /\left(x^{2}+2\right)\right]\) (c) \(\left[x / \sqrt{ \left.\left(x^{2}+2\right)\right]}\right.\) (d) \(\left[1 / \sqrt{ \left.\left(x^{2}+2\right)\right]}\right.\)
See all solutions

Recommended explanations on English Textbooks

Essay Writing Skills

Read Explanation

Essay Prompts

Read Explanation

Research and Composition

Read Explanation

English Grammar

Read Explanation

English Language Study

Read Explanation

Phonetics

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.

Sign-up for free