Question
upstudy study bank question image url

Pattern 2 The common factor Solve for \( A \) if \( A \in\left(-180^{\circ} ; 180^{\circ}\right) \) \[ \sin ^{2} A-\sin A \cos A=0 \]

Ask by Munoz Mann. in South Africa
Mar 11,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The solutions for \( A \) are \( -135^\circ \), \( 0^\circ \), and \( 45^\circ \).

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

The Deep Dive

To solve the equation \( \sin^2 A - \sin A \cos A = 0 \), we can factor out a common term: \[ \sin A (\sin A - \cos A) = 0 \] This gives us two possible cases to consider: 1. \( \sin A = 0 \) 2. \( \sin A - \cos A = 0 \) **Case 1: \( \sin A = 0 \)** The solutions occur at: \[ A = 0^{\circ}, \, 180^{\circ}, \, -180^{\circ} \] In the interval \( (-180^{\circ}, 180^{\circ}) \), we have: \[ A = 0^{\circ} \] **Case 2: \( \sin A - \cos A = 0 \)** This can be rewritten as: \[ \sin A = \cos A \] Dividing both sides by \( \cos A \) (where \( \cos A \neq 0 \)) gives: \[ \tan A = 1 \] The solutions for \( A \) when \( \tan A = 1 \) are: \[ A = 45^{\circ} + k \cdot 180^{\circ}, \quad k \in \mathbb{Z} \] Considering \( k = 0 \) and \( k = -1 \): - \( A = 45^{\circ} \) - \( A = -135^{\circ} \) Now, we compile our solutions: From Case 1: - \( A = 0^{\circ} \) From Case 2: - \( A = 45^{\circ} \) - \( A = -135^{\circ} \) Thus, the solutions for \( A \) in the interval \( (-180^{\circ}, 180^{\circ}) \) are: \[ A = 0^{\circ}, \, 45^{\circ}, \, -135^{\circ} \]

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