Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • Engineering
  • Find the number of solution/solutions of  <img alt="\sin 4\theta \cdot \cos 2\theta =\sin 7\theta \cdot \cos 5\theta" src="https://learn.careers360.com/latex-image/?%5Csin%204%5Ctheta%20%

Find the number of solution/solutions of  \sin 4\theta \cdot \cos 2\theta =\sin 7\theta \cdot \cos 5\theta   \theta \epsilon [0,\frac{\pi}{2}]

Option: 1 1

Option: 2 3

Option: 3 5

Option: 4 7

Answers (1)

best_answer

General Solution of some Standard Equations (Part 1) -

General Solution of some Standard Equations (Part 1)

 

1. sin ? = sin α

\\\text{Given,}\;\sin\theta=\sin\alpha\;\;\Rightarrow \sin\theta-\sin\alpha=0\\\\\mathrm{\Rightarrow 2\cos\frac{\theta+\alpha}{2}\sin\frac{\theta-\alpha}{2}=0}\\\\\mathrm{\Rightarrow \cos\frac{\theta+\alpha}{2}=0\;\;\;\;or\;\;\;\;\sin\frac{\theta-\alpha}{2}=0}\\\\\mathrm{\Rightarrow\frac{\theta+\alpha}{2}=(2n+1)\frac{\pi}{2}\;\;\;\;or\;\;\;\; \frac{\theta-\alpha}{2}=n\pi,\;\;\;n\in\mathbb{I}}\\\\\mathrm{\Rightarrow \theta=(2n+1)\pi-\alpha\;\;\;\;or\;\;\;\;\theta=2n\pi+\alpha,\;\;\;n\in\mathbb{I}}\\\mathrm{\Rightarrow \theta=(any\;odd\;multiple\;of\;\pi)-\alpha\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(i)}\\\mathrm{\Rightarrow \theta=(any\;even\;multiple\;of\;\pi)+\alpha\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(ii)}\\\mathrm{from\;(i)\;and\;(ii)}\\\mathrm{\theta=n\pi+(-1)^n\alpha,\;\;n\in\mathbb{I}}

-

\sin 4\theta \cdot \cos 2\theta =\sin 7\theta \cdot \cos 5\theta \\ 2\sin 4\theta \cdot \cos 2\theta =2\sin 7\theta \cdot \cos 5\theta \\ \sin 6\theta + \sin 2\theta =\sin 12\theta + \sin 2\theta \\ \sin 12\theta=\sin 6\theta\\ 12\theta=n\pi+(-1)^n \cdot 6\theta\\ \text{For n is a odd number}\\ 12\theta=n\pi- 6\theta\\ \theta=\frac{n\pi}{18}\\ \theta\epsilon [0,\frac{\pi}{2}]\\ n=1,3,5,7,9\\ \theta=\{ \frac{\pi}{18},\frac{\pi}{6},\frac{5\pi}{18},\frac{7\pi}{18},\frac{\pi}{2} \}\\ \text{For n is a even number}\\ 12\theta=n\pi+ 6\theta\\ \theta=\frac{n\pi}{6}\\ \theta\epsilon [0,\frac{\pi}{2}] n=0,2\\ \theta=\{0, \frac{\pi}{3} \}\\ \text{Total no. of solutions 7}

Posted by

mansi

View full answer

JEE Main high-scoring chapters and topics

Study 40% syllabus and score up to 100% marks in JEE

Similar Questions

Trending Articles/News

anam-khan

JEE Main 2026: IIT Roorkee’s ECE emerges as top choice after CSE; closing rank rises
anam-khan

JEE Main 2026 application correction begins on jeemain.nta.nic.in; edit details by December 2
anam-khan

JEE Mains 2026 LIVE: NTA to conduct session 1 from January 21 to 30; exam pattern, syllabus

Latest Question