Question
upstudy study bank question image url

Question 4 4.1. Given \( \sin \alpha=-\frac{8}{17} \), where \( 0^{\circ}<\alpha<270^{\circ} \), determine with the aid of a sketch, the value of the following: 4.1.1. \( \tan \alpha \) 4.1.2. \( \cos 2 \alpha \) 4.1.3. \( \sin \frac{\alpha}{2} \) 4.2. Simplify the following trigonometric expression, leaving your answer to only one trigonometric ratio: \( \left[\sin \left(x-1980^{\circ}\right)-\cos (-x)\right]^{2} \)

Ask by Luna Bradley. in South Africa
Mar 18,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

\[ \tan \alpha = \frac{8}{15}, \quad \cos 2\alpha = \frac{161}{289}, \quad \sin\frac{\alpha}{2} = \frac{4\sqrt{17}}{17}, \] \[ \left[ \sin(x-1980^\circ) - \cos(-x) \right]^2 = 1 + \sin 2x. \]

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

Bonus Knowledge

To solve \( \tan \alpha \), we start with the fact that \( \sin \alpha = -\frac{8}{17} \). Since \( \alpha \) is in the third quadrant (where both sine and cosine are negative), we can find \( \cos \alpha \) using the Pythagorean identity: \( \sin^2 \alpha + \cos^2 \alpha = 1 \). This gives us \( \cos \alpha = -\sqrt{1 - \left(-\frac{8}{17}\right)^2} = -\frac{15}{17} \). Therefore, \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} = \frac{-\frac{8}{17}}{-\frac{15}{17}} = \frac{8}{15} \). Next, for \( \cos 2\alpha \), we use the double angle formula: \( \cos 2\alpha = 2\cos^2 \alpha - 1 \). Plugging in our value for \( \cos \alpha \), we find \( \cos 2\alpha = 2\left(-\frac{15}{17}\right)^2 - 1 = 2 \cdot \frac{225}{289} - 1 = \frac{450}{289} - 1 = \frac{450 - 289}{289} = \frac{161}{289} \). For \( \sin \frac{\alpha}{2} \), we use the half-angle formula \( \sin \frac{\alpha}{2} = \sqrt{\frac{1 - \cos \alpha}{2}} \). With \( \cos \alpha = -\frac{15}{17} \), we compute \( \sin \frac{\alpha}{2} = \sqrt{\frac{1 - (-\frac{15}{17})}{2}} = \sqrt{\frac{1 + \frac{15}{17}}{2}} = \sqrt{\frac{\frac{32}{17}}{2}} = \sqrt{\frac{32}{34}} = \sqrt{\frac{16}{17}} = \frac{4}{\sqrt{17}} \). Now for the expression \( \left[\sin \left(x-1980^{\circ}\right)-\cos (-x)\right]^{2} \) simplification: First, recall that \( \cos(-x) = \cos x \). Also, \( 1980^{\circ} \) can be simplified by subtracting \( 1800^{\circ} \) (which is equivalent to \( 5 \times 360^{\circ} \)): \( 1980^{\circ} = 180^{\circ} \). So we have \( \sin(x - 180^{\circ}) = -\sin x \). Therefore, the expression simplifies to \( \left[-\sin x - \cos x\right]^{2} = \left(-\sin x - \cos x\right)^{2} = (\sin x + \cos x)^{2} \). Thus, the final result is \( \sin^2 x + 2\sin x \cos x + \cos^2 x = 1 + \sin(2x) \).

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