To solve the problem of finding [tex]\(\cos 2\theta\)[/tex] and [tex]\(\tan 2\theta\)[/tex] given that [tex]\(\cos \theta = -\frac{8}{17}\)[/tex] and [tex]\(\theta\)[/tex] is an angle in quadrant III, follow these steps:
1. Recall the double angle identities:
- [tex]\(\cos 2\theta = 2\cos^2\theta - 1\)[/tex]
- [tex]\(\tan 2\theta = \frac{2\tan\theta}{1 - \tan^2\theta}\)[/tex]
2. Calculate [tex]\(\cos 2\theta\)[/tex]:
Start by using the double angle formula for cosine:
[tex]\[
\cos 2\theta = 2\cos^2 \theta - 1
\][/tex]
Substitute [tex]\(\cos \theta = -\frac{8}{17}\)[/tex] into the formula:
[tex]\[
\cos^2 \theta = \left(-\frac{8}{17}\right)^2 = \frac{64}{289}
\][/tex]
Then,
[tex]\[
\cos 2\theta = 2 \cdot \frac{64}{289} - 1 = \frac{128}{289} - 1 = \frac{128 - 289}{289} = \frac{-161}{289}
\][/tex]
Therefore,
[tex]\[
\cos 2\theta = -0.5571
\][/tex]
3. Calculate [tex]\(\tan 2\theta\)[/tex]:
First, find [tex]\(\sin \theta\)[/tex]. We know:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
Given [tex]\(\cos^2 \theta = \frac{64}{289}\)[/tex], find [tex]\(\sin^2 \theta\)[/tex]:
[tex]\[
\sin^2 \theta = 1 - \frac{64}{289} = \frac{289 - 64}{289} = \frac{225}{289}
\][/tex]
Therefore,
[tex]\[
\sin \theta = -\sqrt{\frac{225}{289}} = -\frac{15}{17}
\][/tex]
(Negative because [tex]\(\theta\)[/tex] is in the third quadrant where sine is negative.)
Now, calculate [tex]\(\tan \theta\)[/tex]:
[tex]\[
\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-\frac{15}{17}}{-\frac{8}{17}} = \frac{15}{8}
\][/tex]
Use the double angle formula for tangent:
[tex]\[
\tan 2\theta = \frac{2\tan \theta}{1 - \tan^2 \theta} = \frac{2 \cdot \frac{15}{8}}{1 - \left(\frac{15}{8}\right)^2} = \frac{\frac{30}{8}}{1 - \frac{225}{64}} = \frac{\frac{30}{8}}{\frac{64-225}{64}} = \frac{\frac{30}{8}}{\frac{-161}{64}} = \frac{30 \cdot 64}{8 \cdot (-161)} = \frac{1920}{-1288} = \frac{1920}{-1288}
\][/tex]
Simplify further:
[tex]\[
\tan 2\theta = -1.4907
\][/tex]
Therefore, the correct answer should be:
- [tex]\(\cos 2\theta = -0.5571\)[/tex]
- [tex]\(\tan 2\theta = -1.4907\)[/tex]
So, the statement can be completed as:
If [tex]\(\cos \theta = -\frac{8}{17}\)[/tex] and [tex]\(\theta\)[/tex] is in quadrant III, [tex]\(\cos 2\theta = -0.5571\)[/tex] and [tex]\(\tan 2\theta = -1.4907\)[/tex].
