Question
upstudy study bank question image url

en: \( 4 \sin 2 \theta-3 \cos 2 \theta=0 \) Show that \( \tan 2 \theta=\frac{3}{4} \) Hence, solve for \( \theta \) where \( \theta \in\left(-360^{\circ} ;-90^{\circ}\right) \)

Ask by Alexander Pierce. in South Africa
Mar 06,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

\( \theta \approx -71.565^\circ \)

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

Extra Insights

Let's start with the equation given: \[ 4 \sin 2\theta - 3 \cos 2\theta = 0 \] Rearranging this gives: \[ 4 \sin 2\theta = 3 \cos 2\theta \] Dividing both sides by \( \cos 2\theta \) (provided \( \cos 2\theta \neq 0 \)), we have: \[ \frac{4 \sin 2\theta}{\cos 2\theta} = 3 \] This simplifies to: \[ 4 \tan 2\theta = 3 \] Now, isolating \( \tan 2\theta \) results in: \[ \tan 2\theta = \frac{3}{4} \] Now, to solve for \( \theta \) in the interval \( (-360^\circ, -90^\circ) \): 1. The general solution for \( 2\theta \) can be expressed using the arctangent: \[ 2\theta = \tan^{-1}\left(\frac{3}{4}\right) + k \cdot 180^\circ \quad (k \in \mathbb{Z}) \] Calculating \( \tan^{-1}\left(\frac{3}{4}\right) \) gives: \[ 2\theta \approx 36.87^\circ \] So the first solution is: \[ 2\theta \approx 36.87^\circ + k \cdot 180^\circ \] 2. The value of \( \theta \) becomes: For \( k = 0 \): \[ 2\theta \approx 36.87^\circ \Rightarrow \theta \approx 18.44^\circ \] For \( k = 1 \): \[ 2\theta \approx 216.87^\circ \Rightarrow \theta \approx 108.44^\circ \] For \( k = -1 \): \[ 2\theta \approx -143.13^\circ \Rightarrow \theta \approx -71.56^\circ \] 3. The solutions that fit within the interval \( (-360^\circ, -90^\circ) \): - The only valid solution here is: \[ \theta \approx -71.56^\circ \] Therefore, the solution for \( \theta \) is approximately \( -71.56^\circ \) within the specified interval.

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