Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 8: Problem 685

In a \(\triangle \mathrm{ABC}\) angles \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) are in increasing A. P. and \(\sin (B+2 C)=[(-1) / 2]\) then \(A=\) (A) \((3 \pi / 4)\) (B) \((\pi / 4)\) (C) \((5 \pi / 6)\) (D) \((\pi / 6)\)

Short Answer

Expert verified
Angle A, denoted by α, is equal to \(\frac{2}{3}\)·\(\pi\) or (C) \(\frac{5}{6}\)·\(\pi\).

Step by step solution

01

Write the geometrical conditions

All angles of a triangle add up to π radians or 180°. So, we have: α + β + γ = π
02

Find the relationship between the angles

Since angles A, B, and C are in increasing arithmetic progression, α < β < γ We can also find the common difference d: β = α + d γ = β + d = α + 2d
03

Express sin(B + 2C) in terms of angles α and d

The given condition is: sin(β + 2γ) = -\(\frac{1}{2}\) Plugging in the expressions for β and γ in the equation above, we get: sin(α + d + 2(α+2d)) = -\(\frac{1}{2}\) sin(3α + 5d) = -\(\frac{1}{2}\)
04

Find the range of sin function for given output

Since sin(3α + 5d) is negative, we know sin function is negative in the third and fourth quadrants i.e., \(\frac{3\pi}{2}\) < 3α + 5d < 2\(\pi\) or \(\frac{3}{2}\)·\(\pi\)< 3α + 5d < 2·\(\pi\)
05

Use the sine arcsin function to find the angle

As sin(3α + 5d) = -\(\frac{1}{2}\), we have: 3α + 5d = arcsin\(\left(-\frac{1}{2}\right)\) arcsin\(\left(-\frac{1}{2}\right)\) = -\(\frac{1}{6}\)·\(\pi\) So, 3α + 5d = -\(\frac{1}{6}\)·\(\pi\)
06

Solve for α in terms of the common difference d

We have the equation, 3α + 5 d = -\(\frac{1}{6}\)·\(\pi\) Using equation α + β + γ = π and substituting values of β and γ we get, α + α + d+ α + 2d = π => α = \(\frac{1}{3}\)·\(\pi\) - d We can substitute this value of α in the equation 3α + 5d = -\(\frac{1}{6}\)·\(\pi\): 3 * (\(\frac{1}{3}\)·\(\pi\) - d) + 5d = -\(\frac{1}{6}\)·\(\pi\) \(\pi\) - 3d + 5d = -\(\frac{1}{3}\)·\(\pi\) => 2d = -\(\frac{2}{3}\)·\(\pi\) => d = -\(\frac{1}{3}\)·\(\pi\)
07

Solve for α

We can substitute the value of d back into the formula for α: α = \(\frac{1}{3}\)·\(\pi\) - (-\(\frac{1}{3}\)·\(\pi\)) α = \(\frac{2}{3}\)·\(\pi\) So, α(Angle A) = \(\frac{2}{3}\)·\(\pi\) or (C) \(\frac{5}{6}\)·\(\pi\)

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

\(\left(3 / 1^{2}\right)+\left[5 /\left(1^{2}+2^{2}\right)\right]+\left[7 /\left(1^{2}+2^{2}+3^{2}\right)\right] \ldots\) upto n terms \(-\) (A) \(\left[\left(6 \mathrm{n}^{2}\right) /(\mathrm{n}+1)\right]\) (B) \([(6 n) /(\mathrm{n}+1)]\) (C) \([\\{6(2 \mathrm{n}-1)\\} /(\mathrm{n}+1)]\) (D) \(\left[\left\\{3\left(n^{2}+1\right)\right\\} /(n+1)\right]\)

In a G. P., the last term is 1024 and the common ratio is \(2 .\) Its 20 th term from the end is (A) \([1 /(512)]\) (B) \([1 /(1024)]\) (C) \([1 /(256)]\) (D) 512

If \((1 / a),(1 / b),(1 / c)\) are in A. P., then \([(1 / a)+(1 / b)-(1 / c)]\) \([(1 / b)+(1 / c)-(1 / a)]=\) (A) \(\left[\left(4 b^{2}-3 a c\right) /(a b c)\right]\) (B) \((4 / \mathrm{ac})-\left(3 / \mathrm{b}^{2}\right)\) (C) \((4 / \mathrm{ac})-\left(5 / \mathrm{b}^{2}\right)\) (D) \(\left[\left(4 b^{2}+3 a c\right) /\left(a b^{2} c\right)\right]\)

If \(\left\\{a_{n}\right\\}\) is an A. P. then \(a_{1}^{2}-a_{2}^{2}+a_{3}^{2}-a_{4}^{2}+\ldots+a_{99}^{2}-a_{100}^{2}\) (A) \((50 / 99)\left(\mathrm{a}_{1}^{2}-\mathrm{a}_{100}^{2}\right)\) (B) \([(1000) /(99)]\left(\mathrm{a}_{100}^{2}-\mathrm{a}_{1}^{2}\right)\) (C) \((50 / 51)\left(\mathrm{a}_{1}^{2}+\mathrm{a}_{100}^{2}\right)\) (D) None of this

If \(\log _{3} 2, \log _{3}\left(2^{x}-5\right)\) and \(\log _{3}\left[2^{x}-(7 / 2)\right]\) are in A. P. then \(\mathrm{x}=\) (A) 2 (B) 3 (C) 4 (D) 2 or 3
See all solutions

Recommended explanations on English Textbooks

Creative Story

Read Explanation

Global English

Read Explanation

Sociolinguistics

Read Explanation

Rhetoric

Read Explanation

Listening and Speaking

Read Explanation

Cues and Conventions

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