Use the information given about the angle $$ heta, \cot heta=-4, \sec heta

Question

Use the information given about the angle $$ 	heta, \cot 	heta=-4, \sec 	heta<0,0 \leq 	heta<2 \pi $$, to find the exact values of the following. (a) $$ \sin (2 	heta) $$, (b) $$ \cos (2 	heta) $$, (c) $$ \sin \frac{	heta}{2} $$, and (d) $$ \cos \frac{	heta}{2} $$ (a) $$ \sin (2 	heta)=-\frac{8}{17} $$ (Type an exact answer, using radicals as needed.) (b) $$ \cos (2 	heta)=\frac{15}{17} $$ (Type an exact answer, using radicals as needed.) (c) $$ \sin \frac{	heta}{2}=\square $$ (Type an exact answer, using radicals as needed.)

UpStudy AI · Accepted Answer

To solve the problem, we start with the given information about the angle $$ \theta $$:

1. $$ \cot \theta = -4 $$
2. $$ \sec \theta < 0 $$
3. $$ 0 \leq \theta < 2\pi $$

### Step 1: Determine the quadrant of $$ \theta $$

Since $$ \cot \theta = -4 $$, this means that $$ \tan \theta = -\frac{1}{4} $$. The cotangent is negative in the second and fourth quadrants. However, since $$ \sec \theta < 0 $$, which implies that $$ \cos \theta < 0 $$, we conclude that $$ \theta $$ must be in the second quadrant.

### Step 2: Find $$ \sin \theta $$ and $$ \cos \theta $$

Using the identity $$ \cot \theta = \frac{\cos \theta}{\sin \theta} $$, we can express $$ \cos \theta $$ and $$ \sin \theta $$ in terms of a right triangle. Let:

- $$ \cos \theta = -4k $$
- $$ \sin \theta = k $$

From the Pythagorean identity $$ \sin^2 \theta + \cos^2 \theta = 1 $$:

$$
k^2 + (-4k)^2 = 1
$$
$$
k^2 + 16k^2 = 1
$$
$$
17k^2 = 1
$$
$$
k^2 = \frac{1}{17}
$$
$$
k = \frac{1}{\sqrt{17}} \quad (\text{since } k > 0 \text{ in the second quadrant})
$$

Thus, we have:

$$
\sin \theta = k = \frac{1}{\sqrt{17}}, \quad \cos \theta = -4k = -\frac{4}{\sqrt{17}}
$$

### Step 3: Calculate $$ \sin(2\theta) $$ and $$ \cos(2\theta) $$

Using the double angle formulas:

$$
\sin(2\theta) = 2 \sin \theta \cos \theta
$$
$$
\cos(2\theta) = \cos^2 \theta - \sin^2 \theta
$$

Calculating $$ \sin(2\theta) $$:

$$
\sin(2\theta) = 2 \left(\frac{1}{\sqrt{17}}\right) \left(-\frac{4}{\sqrt{17}}\right) = -\frac{8}{17}
$$

Calculating $$ \cos(2\theta) $$:

$$
\cos(2\theta) = \left(-\frac{4}{\sqrt{17}}\right)^2 - \left(\frac{1}{\sqrt{17}}\right)^2 = \frac{16}{17} - \frac{1}{17} = \frac{15}{17}
$$

### Step 4: Calculate $$ \sin\left(\frac{\theta}{2}\right) $$ and $$ \cos\left(\frac{\theta}{2}\right) $$

Using the half-angle formulas:

$$
\sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos \theta}{2}}
$$
$$
\cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos \theta}{2}}
$$

Calculating $$ \sin\left(\frac{\theta}{2}\right) $$:

$$
\sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \left(-\frac{4}{\sqrt{17}}\right)}{2}} = \sqrt{\frac{1 + \frac{4}{\sqrt{17}}}{2}} = \sqrt{\frac{\frac{\sqrt{17} + 4}{\sqrt{17}}}{2}} = \sqrt{\frac{\sqrt{17} + 4}{2\sqrt{17}}}
$$

Calculating $$ \cos\left(\frac{\theta}{2}\right) $$:

$$
\cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \left(-\frac{4}{\sqrt{17}}\right)}{2}} = \sqrt{\frac{1 - \frac{4}{\sqrt{17}}}{2}} = \sqrt{\frac{\frac{\sqrt{17} - 4}{\sqrt{17}}}{2}} = \sqrt{\frac{\sqrt{17} - 4}{2\sqrt{17}}}
$$

### Final Answers

(a) $$ \sin(2\theta) = -\frac{8}{17} $$

(b) $$ \cos(2\theta) = \frac{15}{17} $$

(c) $$ \sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{\sqrt{17} + 4}{2\sqrt{17}}} $$

(d) $$ \cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{\sqrt{17} - 4}{2\sqrt{17}}} $$
(a) $$ \sin(2\theta) = -\frac{8}{17} $$ (b) $$ \cos(2\theta) = \frac{15}{17} $$ (c) $$ \sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{\sqrt{17} + 4}{2\sqrt{17}}} $$ (d) $$ \cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{\sqrt{17} - 4}{2\sqrt{17}}} $$

Use the information given about the angle , to find the exact values of the following.
(a) , (b) , © , and (d)
(a) (Type an exact answer, using radicals as needed.)
(b) (Type an exact answer, using radicals as needed.)
© (Type an exact answer, using radicals as needed.)

Upstudy AI Solution

Answer

Solution

Step 1: Determine the quadrant of

Step 2: Find and

Step 3: Calculate and

Step 4: Calculate and

Final Answers

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Use the information given about the angle , to find the exact values of the following. (a) , (b) , © , and (d) (a) (Type an exact answer, using radicals as needed.) (b) (Type an exact answer, using radicals as needed.) © (Type an exact answer, using radicals as needed.)