Question
upstudy study bank question image url

2.15 Use \( \cos (\alpha-\beta) \) to derive a formula for \( \sin (\alpha-\beta) \). (Hint: use co-function) \( \qquad \) \( \qquad \) (3) 21.6 Use \( \cos (\alpha-\beta) \) to derive a formula for \( \sin (\alpha+\beta) \). (Hint: use ce-function) \( \qquad \) \( \qquad \) (3) [22] Cond 12 Mathenutics SBA 2005 Prge 18 of 33

Ask by Henry Olson. in South Africa
Mar 09,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

**Deriving Formulas for \( \sin(\alpha \pm \beta) \)** 1. **For \( \sin(\alpha - \beta) \):** \[ \sin(\alpha - \beta) = \sin\alpha \cos\beta - \cos\alpha \sin\beta \] 2. **For \( \sin(\alpha + \beta) \):** \[ \sin(\alpha + \beta) = \sin\alpha \cos\beta + \cos\alpha \sin\beta \]

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

To derive a formula for \( \sin (\alpha-\beta) \) using \( \cos (\alpha-\beta) \), recall that \( \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \). Using the co-function identity, we know that \( \sin(\theta) = \cos\left(\frac{\pi}{2} - \theta\right) \). By exploiting that relationship, we can derive the sine difference formula: \[ \sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta \] Now for \( \sin(\alpha+\beta) \), we can manipulate the expression for \( \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \). Again using co-function identities, and knowing that \( \sin(\alpha+\beta) = \cos\left(\frac{\pi}{2} - (\alpha+\beta)\right) \), we can derive the sum of angles as: \[ \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \]

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