Westonci.ca is your trusted source for finding answers to a wide range of questions, backed by a knowledgeable community. Get detailed and accurate answers to your questions from a community of experts on our comprehensive Q&A platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
Sure, let's solve this problem step by step.
Given:
[tex]\[ \sin \theta = -\frac{7}{25} \][/tex]
and we know that [tex]\( \theta \)[/tex] lies in the third quadrant, that is,
[tex]\[ \pi < \theta < \frac{3\pi}{2} \][/tex]
### Step 1: Determine [tex]\(\cos \theta\)[/tex]
First, recall the Pythagorean identity:
[tex]\[ \sin^2 \theta + \cos^2 \theta = 1 \][/tex]
We are given [tex]\(\sin \theta = -\frac{7}{25}\)[/tex], so:
[tex]\[ \left( -\frac{7}{25} \right)^2 + \cos^2 \theta = 1 \][/tex]
[tex]\[ \left( \frac{49}{625} \right) + \cos^2 \theta = 1 \][/tex]
[tex]\[ \cos^2 \theta = 1 - \frac{49}{625} \][/tex]
[tex]\[ \cos^2 \theta = \frac{625}{625} - \frac{49}{625} \][/tex]
[tex]\[ \cos^2 \theta = \frac{576}{625} \][/tex]
[tex]\[ \cos \theta = \pm \sqrt{\frac{576}{625}} \][/tex]
[tex]\[ \cos \theta = \pm \frac{24}{25} \][/tex]
Since [tex]\(\theta\)[/tex] lies in the third quadrant, both sine and cosine are negative:
[tex]\[ \cos \theta = -\frac{24}{25} \][/tex]
### Step 2: Use the double-angle identity for sine
The double-angle identity for sine is:
[tex]\[ \sin 2\theta = 2 \sin \theta \cos \theta \][/tex]
Substitute the values we have:
[tex]\[ \sin 2\theta = 2 \left( -\frac{7}{25} \right) \left( -\frac{24}{25} \right) \][/tex]
[tex]\[ \sin 2\theta = 2 \cdot \frac{7 \cdot 24}{25 \cdot 25} \][/tex]
[tex]\[ \sin 2\theta = 2 \cdot \frac{168}{625} \][/tex]
[tex]\[ \sin 2\theta = \frac{336}{625} \][/tex]
So, the exact solution for [tex]\(\sin 2 \theta\)[/tex] is:
[tex]\[ \frac{336}{625} \][/tex]
Thus, the correct answer is:
[tex]\[ \frac{336}{625} \][/tex]
Given:
[tex]\[ \sin \theta = -\frac{7}{25} \][/tex]
and we know that [tex]\( \theta \)[/tex] lies in the third quadrant, that is,
[tex]\[ \pi < \theta < \frac{3\pi}{2} \][/tex]
### Step 1: Determine [tex]\(\cos \theta\)[/tex]
First, recall the Pythagorean identity:
[tex]\[ \sin^2 \theta + \cos^2 \theta = 1 \][/tex]
We are given [tex]\(\sin \theta = -\frac{7}{25}\)[/tex], so:
[tex]\[ \left( -\frac{7}{25} \right)^2 + \cos^2 \theta = 1 \][/tex]
[tex]\[ \left( \frac{49}{625} \right) + \cos^2 \theta = 1 \][/tex]
[tex]\[ \cos^2 \theta = 1 - \frac{49}{625} \][/tex]
[tex]\[ \cos^2 \theta = \frac{625}{625} - \frac{49}{625} \][/tex]
[tex]\[ \cos^2 \theta = \frac{576}{625} \][/tex]
[tex]\[ \cos \theta = \pm \sqrt{\frac{576}{625}} \][/tex]
[tex]\[ \cos \theta = \pm \frac{24}{25} \][/tex]
Since [tex]\(\theta\)[/tex] lies in the third quadrant, both sine and cosine are negative:
[tex]\[ \cos \theta = -\frac{24}{25} \][/tex]
### Step 2: Use the double-angle identity for sine
The double-angle identity for sine is:
[tex]\[ \sin 2\theta = 2 \sin \theta \cos \theta \][/tex]
Substitute the values we have:
[tex]\[ \sin 2\theta = 2 \left( -\frac{7}{25} \right) \left( -\frac{24}{25} \right) \][/tex]
[tex]\[ \sin 2\theta = 2 \cdot \frac{7 \cdot 24}{25 \cdot 25} \][/tex]
[tex]\[ \sin 2\theta = 2 \cdot \frac{168}{625} \][/tex]
[tex]\[ \sin 2\theta = \frac{336}{625} \][/tex]
So, the exact solution for [tex]\(\sin 2 \theta\)[/tex] is:
[tex]\[ \frac{336}{625} \][/tex]
Thus, the correct answer is:
[tex]\[ \frac{336}{625} \][/tex]
Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.