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To find the exact values of [tex]\(\cos \frac{\theta}{2}\)[/tex] and [tex]\(\tan \frac{\theta}{2}\)[/tex] given that [tex]\(\cos \theta = -\frac{3}{\sqrt{10}}\)[/tex] and [tex]\(\frac{\pi}{2} < \theta < \pi\)[/tex], we can use half-angle formulas. Here’s a detailed, step-by-step solution:
### Step 1: Identify the given information
We are given that:
[tex]\[ \cos \theta = -\frac{3}{\sqrt{10}} \][/tex]
and the angle [tex]\(\theta\)[/tex] lies in the interval [tex]\(\frac{\pi}{2} < \theta < \pi\)[/tex].
### Step 2: Recall the half-angle formulas
The half-angle formulas for cosine and tangent are:
[tex]\[ \cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}} \][/tex]
[tex]\[ \tan \frac{\theta}{2} = \frac{\sin \theta}{1 + \cos \theta} \][/tex]
### Step 3: Determine the signs of [tex]\(\cos \frac{\theta}{2}\)[/tex] and [tex]\(\tan \frac{\theta}{2}\)[/tex]
Since [tex]\(\frac{\pi}{4} < \frac{\theta}{2} < \frac{\pi}{2}\)[/tex], both [tex]\(\cos \frac{\theta}{2}\)[/tex] and [tex]\(\tan \frac{\theta}{2}\)[/tex] are positive in this range.
### Step 4: Compute [tex]\(\cos \frac{\theta}{2}\)[/tex]
Use the half-angle formula for cosine with the correct sign:
[tex]\[ \cos \frac{\theta}{2} = \sqrt{\frac{1 + \cos \theta}{2}} \][/tex]
Substitute the given value of [tex]\(\cos \theta\)[/tex]:
[tex]\[ \cos \frac{\theta}{2} = \sqrt{\frac{1 + \left(-\frac{3}{\sqrt{10}}\right)}{2}} \][/tex]
[tex]\[ \cos \frac{\theta}{2} = \sqrt{\frac{1 - \frac{3}{\sqrt{10}}}{2}} \][/tex]
[tex]\[ \cos \frac{\theta}{2} = \sqrt{\frac{2 - \frac{3}{\sqrt{10}}}{4}} \][/tex]
[tex]\[ \cos \frac{\theta}{2} = \sqrt{\frac{2\sqrt{10} - 3}{4\sqrt{10}}} \][/tex]
Evaluating this expression gives:
[tex]\[ \cos \frac{\theta}{2} \approx 0.16018224300696726 \][/tex]
### Step 5: Compute [tex]\(\tan \frac{\theta}{2}\)[/tex]
We first need [tex]\(\sin \theta\)[/tex]. Using the Pythagorean identity [tex]\(\sin^2 \theta + \cos^2 \theta = 1\)[/tex]:
[tex]\[ \sin \theta = \sqrt{1 - \cos^2 \theta} \][/tex]
[tex]\[ \sin \theta = \sqrt{1 - \left(-\frac{3}{\sqrt{10}}\right)^2} \][/tex]
[tex]\[ \sin \theta = \sqrt{1 - \frac{9}{10}} \][/tex]
[tex]\[ \sin \theta = \sqrt{\frac{1}{10}} \][/tex]
[tex]\[ \sin \theta = \frac{1}{\sqrt{10}} \][/tex]
Now use the half-angle formula for tangent:
[tex]\[ \tan \frac{\theta}{2} = \frac{\sin \theta}{1 + \cos \theta} \][/tex]
Substitute the values of [tex]\(\sin \theta\)[/tex] and [tex]\(\cos \theta\)[/tex]:
[tex]\[ \tan \frac{\theta}{2} = \frac{\frac{1}{\sqrt{10}}}{1 - \frac{3}{\sqrt{10}}} \][/tex]
[tex]\[ \tan \frac{\theta}{2} = \frac{\frac{1}{\sqrt{10}}}{\frac{\sqrt{10} - 3}{\sqrt{10}}} \][/tex]
[tex]\[ \tan \frac{\theta}{2} = \frac{1}{\sqrt{10} - 3} \][/tex]
Evaluating this expression gives:
[tex]\[ \tan \frac{\theta}{2} \approx 6.162277660168378 \][/tex]
So the exact values are:
[tex]\[ \cos \frac{\theta}{2} \approx 0.16018224300696726 \][/tex]
[tex]\[ \tan \frac{\theta}{2} \approx 6.162277660168378 \][/tex]
### Step 1: Identify the given information
We are given that:
[tex]\[ \cos \theta = -\frac{3}{\sqrt{10}} \][/tex]
and the angle [tex]\(\theta\)[/tex] lies in the interval [tex]\(\frac{\pi}{2} < \theta < \pi\)[/tex].
### Step 2: Recall the half-angle formulas
The half-angle formulas for cosine and tangent are:
[tex]\[ \cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}} \][/tex]
[tex]\[ \tan \frac{\theta}{2} = \frac{\sin \theta}{1 + \cos \theta} \][/tex]
### Step 3: Determine the signs of [tex]\(\cos \frac{\theta}{2}\)[/tex] and [tex]\(\tan \frac{\theta}{2}\)[/tex]
Since [tex]\(\frac{\pi}{4} < \frac{\theta}{2} < \frac{\pi}{2}\)[/tex], both [tex]\(\cos \frac{\theta}{2}\)[/tex] and [tex]\(\tan \frac{\theta}{2}\)[/tex] are positive in this range.
### Step 4: Compute [tex]\(\cos \frac{\theta}{2}\)[/tex]
Use the half-angle formula for cosine with the correct sign:
[tex]\[ \cos \frac{\theta}{2} = \sqrt{\frac{1 + \cos \theta}{2}} \][/tex]
Substitute the given value of [tex]\(\cos \theta\)[/tex]:
[tex]\[ \cos \frac{\theta}{2} = \sqrt{\frac{1 + \left(-\frac{3}{\sqrt{10}}\right)}{2}} \][/tex]
[tex]\[ \cos \frac{\theta}{2} = \sqrt{\frac{1 - \frac{3}{\sqrt{10}}}{2}} \][/tex]
[tex]\[ \cos \frac{\theta}{2} = \sqrt{\frac{2 - \frac{3}{\sqrt{10}}}{4}} \][/tex]
[tex]\[ \cos \frac{\theta}{2} = \sqrt{\frac{2\sqrt{10} - 3}{4\sqrt{10}}} \][/tex]
Evaluating this expression gives:
[tex]\[ \cos \frac{\theta}{2} \approx 0.16018224300696726 \][/tex]
### Step 5: Compute [tex]\(\tan \frac{\theta}{2}\)[/tex]
We first need [tex]\(\sin \theta\)[/tex]. Using the Pythagorean identity [tex]\(\sin^2 \theta + \cos^2 \theta = 1\)[/tex]:
[tex]\[ \sin \theta = \sqrt{1 - \cos^2 \theta} \][/tex]
[tex]\[ \sin \theta = \sqrt{1 - \left(-\frac{3}{\sqrt{10}}\right)^2} \][/tex]
[tex]\[ \sin \theta = \sqrt{1 - \frac{9}{10}} \][/tex]
[tex]\[ \sin \theta = \sqrt{\frac{1}{10}} \][/tex]
[tex]\[ \sin \theta = \frac{1}{\sqrt{10}} \][/tex]
Now use the half-angle formula for tangent:
[tex]\[ \tan \frac{\theta}{2} = \frac{\sin \theta}{1 + \cos \theta} \][/tex]
Substitute the values of [tex]\(\sin \theta\)[/tex] and [tex]\(\cos \theta\)[/tex]:
[tex]\[ \tan \frac{\theta}{2} = \frac{\frac{1}{\sqrt{10}}}{1 - \frac{3}{\sqrt{10}}} \][/tex]
[tex]\[ \tan \frac{\theta}{2} = \frac{\frac{1}{\sqrt{10}}}{\frac{\sqrt{10} - 3}{\sqrt{10}}} \][/tex]
[tex]\[ \tan \frac{\theta}{2} = \frac{1}{\sqrt{10} - 3} \][/tex]
Evaluating this expression gives:
[tex]\[ \tan \frac{\theta}{2} \approx 6.162277660168378 \][/tex]
So the exact values are:
[tex]\[ \cos \frac{\theta}{2} \approx 0.16018224300696726 \][/tex]
[tex]\[ \tan \frac{\theta}{2} \approx 6.162277660168378 \][/tex]
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