Question
upstudy study bank question image url

\( \sin \theta=-\frac{\sqrt{3}}{3}, \frac{3 \pi}{2}<\theta<2 \pi \) Find \( \sin (2 \theta) \)

Ask by Vargas Bartlett. in the United States
Mar 14,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

\[ \sin(2\theta) = -\frac{2\sqrt{2}}{3} \]

Solution

Sign in to Unlock Answers for Free!

A Learning Platform Trusted by Millions of Real Students and Teachers.

star-icon Unlock

Answered by UpStudy AI and reviewed by a Professional Tutor

The Deep Dive

To find \( \sin(2\theta) \), we can first determine the angle \( \theta \) using the given sine value. The sine function is negative in the fourth quadrant, and we know that: \[ \sin \theta = -\frac{\sqrt{3}}{3} \] The reference angle where \( \sin \theta = \frac{\sqrt{3}}{3} \) is \( \theta = \frac{\pi}{6} \). Therefore, in the fourth quadrant, we can express the angle \( \theta \) as: \[ \theta = 2\pi - \frac{\pi}{6} = \frac{12\pi}{6} - \frac{\pi}{6} = \frac{11\pi}{6} \] Now that we have \( \theta = \frac{11\pi}{6} \), we can use the double angle formula for sine: \[ \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \] We already have \( \sin \theta = -\frac{\sqrt{3}}{3} \). Next, we need to find \( \cos \theta \). Using the Pythagorean identity: \[ \sin^2 \theta + \cos^2 \theta = 1 \] Substituting the known value of \( \sin \theta \): \[ \left(-\frac{\sqrt{3}}{3}\right)^2 + \cos^2 \theta = 1 \] \[ \frac{3}{9} + \cos^2 \theta = 1 \quad \Rightarrow \quad \cos^2 \theta = 1 - \frac{1}{3} = \frac{2}{3} \] Taking the square root gives: \[ \cos \theta = \sqrt{\frac{2}{3}} \quad \text{(since} \ \cos \theta > 0 \ \text{in the fourth quadrant)} \] Now, substituting back into the double angle formula: \[ \sin(2\theta) = 2 \left(-\frac{\sqrt{3}}{3}\right) \left(\frac{\sqrt{2}}{\sqrt{3}}\right) \] Simplifying gives: \[ \sin(2\theta) = 2 \left(-\frac{\sqrt{3}}{3}\right) \left(\frac{\sqrt{2}}{\sqrt{3}}\right) = -\frac{2\sqrt{6}}{9} \] Thus, the final answer is: \[ \sin(2\theta) = -\frac{2\sqrt{6}}{9} \]

Related Questions

Latest Trigonometry Questions

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy