Given that \(\tan \theta = -\frac{7}{4}\) and \(\frac{\pi}{2} < \theta < \pi\), we will find the exact values of the trigonometric functions \( \sin 2\theta\), \( \csc 2\theta\), \( \cos 2\theta\), \( \sec 2\theta\), \( \tan 2\theta\), and \( \cot 2\theta\).
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Finding \( \sin \theta \) and \( \cos \theta \)
We know that: \[ \tan \theta = \frac{\sin \theta}{\cos \theta} \] Here, \(\tan \theta = -\frac{7}{4}\), which gives us: \[ \sin \theta = -7k, \quad \cos \theta = 4k \] for some \(k\). Since we know \(\theta\) is in the second quadrant, \(\sin \theta > 0\) and \(\cos \theta < 0\), meaning we need to choose \(k\) such that: \[ \sin \theta = 7k \quad \text{and} \quad \cos \theta = -4k \]
We can now use the Pythagorean identity: \[ \sin^2 \theta + \cos^2 \theta = 1 \] Substituting \( \sin \theta\) and \( \cos \theta \) gives: \[ (7k)^2 + (-4k)^2 = 1 \] \[ 49k^2 + 16k^2 = 1 \] \[ 65k^2 = 1 \Rightarrow k^2 = \frac{1}{65} \Rightarrow k = \frac{1}{\sqrt{65}} \quad (k > 0) \]
Now we can find \( \sin \theta\) and \( \cos \theta \): \[ \sin \theta = 7k = \frac{7}{\sqrt{65}}, \quad \cos \theta = -4k = -\frac{4}{\sqrt{65}} \]
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Finding \( \sin 2\theta\) and \( \cos 2\theta\)
Using the double angle formulas: \[ \sin 2\theta = 2 \sin \theta \cos \theta \] \[ \sin 2\theta = 2 \left(\frac{7}{\sqrt{65}}\right) \left(-\frac{4}{\sqrt{65}}\right) = -\frac{56}{65} \]
For \( \cos 2\theta\): \[ \cos 2\theta = \cos^2 \theta - \sin^2 \theta \] \[ \cos^2 \theta = \left(-\frac{4}{\sqrt{65}}\right)^2 = \frac{16}{65}, \quad \sin^2 \theta = \left(\frac{7}{\sqrt{65}}\right)^2 = \frac{49}{65} \] Thus: \[ \cos 2\theta = \frac{16}{65} - \frac{49}{65} = -\frac{33}{65} \]
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Finding \( \csc 2\theta \), \( \sec 2\theta \), \( \tan 2\theta \), and \( \cot 2\theta \)
Now we compute the rest of the functions using the identities for cosecant, secant, tangent, and cotangent: \[ \csc 2\theta = \frac{1}{\sin 2\theta} = -\frac{65}{56} \]
\[ \sec 2\theta = \frac{1}{\cos 2\theta} = -\frac{65}{33} \]
\[ \tan 2\theta = \frac{\sin 2\theta}{\cos 2\theta} = \frac{-\frac{56}{65}}{-\frac{33}{65}} = \frac{56}{33} \]
\[ \cot 2\theta = \frac{1}{\tan 2\theta} = \frac{33}{56} \]
Putting it all together, we have:
\[ \sin 2\theta = -\frac{56}{65} \] \[ \csc 2\theta = -\frac{65}{56} \] \[ \cos 2\theta = -\frac{33}{65} \] \[ \sec 2\theta = -\frac{65}{33} \] \[ \tan 2\theta = \frac{56}{33} \] \[ \cot 2\theta = \frac{33}{56} \]