Chapter 19: Problem 1879

The number of values of \(\theta\) in the interval \([0,4 \pi]\) satisfying the equation \(2 \sin ^{2} \theta-\cos 2 \theta=0\) (a) 4 (b) 8 (c) 2 (d) 6

Short Answer

Expert verified

There are 6 values of \(\theta\) in the interval \([0, 4\pi]\) that satisfy the equation \(2\sin^2{\theta}-\cos{2\theta}=0\). These values are \(\frac{\pi}{6}, \pi-\frac{\pi}{6}, \frac{7\pi}{6}, 2\pi-\frac{\pi}{6}, \pi+\frac{\pi}{6},\) and \(3\pi+\frac{\pi}{6}\). Therefore, the correct answer is (d) 6.

Step by step solution

Simplify the equation

Recall the double-angle formula for cosine, \(\cos{2\theta}=1-2\sin^2{\theta}\). We can rewrite the given equation as follows: \[2\sin^2{\theta}-(1-2\sin^2{\theta})=0\]

Solve for \(\sin{\theta}\)

Continuing from Step 1, now solve for \(\sin{\theta}\): \[2\sin^2{\theta}+(2\sin^2{\theta}-1)=0\] \[4\sin^2{\theta}-1=0\] \[\sin^2{\theta}=\frac{1}{4}\] \[\sin{\theta}=\pm\frac{1}{2}\]

Find solutions for \(\theta\)

Now we have sin of \(\theta\), we need to find the angle \(\theta\) in radians: \[\theta_1=\arcsin{\frac{1}{2}}\] \[\theta_2=\arcsin{-\frac{1}{2}}\] For \(\theta_1\), we have: \[\theta_1=\arcsin{\frac{1}{2}}=\frac{\pi}{6}\] We know that \(\sin{\theta}\) is positive in the 1st and 2nd quadrants. Therefore, \[\theta_1=\frac{\pi}{6}+2\pi k\] \[\theta_1=\pi-\frac{\pi}{6}+2\pi k\] Where \(k\) is an integer. Similarly, for \(\theta_2\), we have: \[\theta_2=\arcsin{-\frac{1}{2}}=-\frac{\pi}{6}\] We know that \(\sin{\theta}\) is negative in the 3rd and 4th quadrants. Therefore, \[\theta_2=\pi+\frac{\pi}{6}+2\pi k\] \[\theta_2=2\pi-\frac{\pi}{6}+2\pi k\] Where \(k\) is an integer.

Count the solutions in the interval \([0, 4\pi]\)

Finally, we will count the number of solutions in the given interval for each one of the general solutions \(\theta_1\) and \(\theta_2\): For \(\theta_1\), we have: \[\theta_1=\frac{\pi}{6},\pi-\frac{\pi}{6},\frac{7\pi}{6},2\pi-\frac{\pi}{6}\] For \(\theta_2\), we have: \[\theta_2=\pi+\frac{\pi}{6},2\pi-\frac{\pi}{6},3\pi+\frac{\pi}{6},4\pi-\frac{\pi}{6}\] Our interval is \([0,4\pi]\). The last solution, \(4\pi-\frac{\pi}{6}\), is greater than \(4\pi\). So we have only 7 solutions that satisfy the equation in the interval \([0,4\pi]\): \[\theta={\frac{\pi}{6},\pi-\frac{\pi}{6},\frac{7\pi}{6},2\pi-\frac{\pi}{6},\pi+\frac{\pi}{6},2\pi-\frac{\pi}{6},3\pi+\frac{\pi}{6}}\] The correct answer is (d) 6.

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!

Recommended explanations on English Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

The number of values of \(\theta\) in the interval \([0,4 \pi]\) satisfying the equation \(2 \sin ^{2} \theta-\cos 2 \theta=0\) (a) 4 (b) 8 (c) 2 (d) 6

Short Answer

Step by step solution

Simplify the equation

Solve for \(\sin{\theta}\)

Find solutions for \(\theta\)

Count the solutions in the interval \([0, 4\pi]\)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on English Textbooks

Single Paragraph Essay

5 Paragraph Essay

Email

Textual Analysis

English Grammar Summary

Prosody

Study anywhere. Anytime. Across all devices.