Use the information given about the angle $$ heta, \cot heta=-8, \sec heta

Question

Use the information given about the angle $$ 	heta, \cot 	heta=-8, \sec 	heta<0,0 \leq 	heta<2 \pi $$, to find the exact values of the following. $$ \begin{array}{l}	ext { (a) } \sin (2 	heta), 	ext { (b) } \cos (2 	heta), 	ext { (c) } \sin \frac{	heta}{2} 	ext {, and (d) } \cos \frac{	heta}{2} \ 	ext { (a) } \sin (2 	heta)=\square 	ext { (Type an exact answer, using radicals as needed.) }\end{array} $$

UpStudy AI · Accepted Answer

Given:
- $$ \cot \theta = -8 $$
- $$ \sec \theta < 0 $$
- $$ 0 \leq \theta < 2\pi $$

We need to find $$ \sin(2\theta) $$.

**Step 1: Determine $$ \sin \theta $$ and $$ \cos \theta $$**

From $$ \cot \theta = \frac{\cos \theta}{\sin \theta} = -8 $$, we can express:
$$ \cos \theta = -8 \sin \theta $$

Since $$ \sec \theta < 0 $$, it implies $$ \cos \theta < 0 $$.

Using the Pythagorean identity:
$$ \sin^2 \theta + \cos^2 \theta = 1 $$
Substitute $$ \cos \theta = -8 \sin \theta $$:
$$ \sin^2 \theta + (-8 \sin \theta)^2 = 1 $$
$$ \sin^2 \theta + 64 \sin^2 \theta = 1 $$
$$ 65 \sin^2 \theta = 1 $$
$$ \sin \theta = \frac{1}{\sqrt{65}} $$
$$ \cos \theta = -8 \cdot \frac{1}{\sqrt{65}} = -\frac{8}{\sqrt{65}} $$

**Step 2: Compute $$ \sin(2\theta) $$**

Using the double-angle identity:
$$ \sin(2\theta) = 2 \sin \theta \cos \theta $$
$$ \sin(2\theta) = 2 \cdot \frac{1}{\sqrt{65}} \cdot \left(-\frac{8}{\sqrt{65}}\right) $$
$$ \sin(2\theta) = \frac{-16}{65} $$

**Answer:**
$$ \sin(2\theta) = -\frac{16}{65} $$

Answer:
Problem a Answer

$$\sin(2\theta) = -\dfrac{16}{65}$$
$$ \sin(2\theta) = -\frac{16}{65} $$

Use the information given about the angle , to find the exact values of the following.

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