Question
upstudy study bank question image url

QUESTION 4 1.1 Given that \( \cos 42^{\circ}=k \), determine the following in terms of \( k \). 4.1.1 \( \sin 42^{\circ} \) (3) 4.1.2 \( \cos 21^{\circ} \) (3) Simplify the following expression to a single trigonometric ratio. \[ \frac{\cos (-\theta) \cdot \sin \left(90^{\circ}+\theta\right)}{\cos \left(360^{\circ}-\theta\right) \cdot \sin ^{2}\left(540^{\circ}+\theta\right)} \]

Ask by Norris Mccarthy. in South Africa
Mar 17,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

\[ \sin 42^\circ = \sqrt{1 - k^2}, \quad \cos 21^\circ = \sqrt{\frac{k + 1}{2}}, \quad \frac{\cos (-\theta) \cdot \sin \left(90^\circ + \theta\right)}{\cos \left(360^\circ - \theta\right) \cdot \sin^2 \left(540^\circ + \theta\right)} = \frac{\cos \theta}{\sin^2 \theta} \]

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 solve for \( \sin 42^{\circ} \) when \( \cos 42^{\circ} = k \), we can use the Pythagorean identity, \( \sin^2 \theta + \cos^2 \theta = 1 \). Thus: \[ \sin 42^{\circ} = \sqrt{1 - \cos^2 42^{\circ}} = \sqrt{1 - k^2} \] For \( \cos 21^{\circ} \), we can make use of the double angle formula: \[ \cos 21^{\circ} = \cos(42/2) = \sqrt{\frac{1 + \cos 42^{\circ}}{2}} = \sqrt{\frac{1 + k}{2}} \] Now, let's simplify the given expression: \[ \frac{\cos (-\theta) \cdot \sin(90^{\circ} + \theta)}{\cos(360^{\circ} - \theta) \cdot \sin^{2}(540^{\circ} + \theta)} \] First, we know that \( \cos(-\theta) = \cos(\theta) \) and \( \sin(90^{\circ} + \theta) = \cos \theta \). Thus, the numerator becomes: \[ \cos(\theta) \cdot \cos(\theta) = \cos^2(\theta) \] For the denominator, we have: \[ \cos(360^{\circ} - \theta) = \cos(\theta) \] And \( \sin(540^{\circ} + \theta) = \sin(180^{\circ} + \theta) = -\sin(\theta) \), so: \[ \sin^2(540^{\circ} + \theta) = \sin^2(\theta) \] Now the denominator becomes: \[ \cos(\theta) \cdot \sin^2(\theta) \] Putting it all together: \[ \frac{\cos^2(\theta)}{\cos(\theta) \cdot \sin^2(\theta)} = \frac{\cos(\theta)}{\sin^2(\theta)} = \cot(\theta) \cdot \csc(\theta) \] The final simplified expression is: \[ \cot(\theta) \cdot \csc(\theta) \]

Related 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