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To solve for \(\sin 2\theta\), \(\cos 2\theta\), and \(\tan 2\theta\) given that \(\sin \theta = -\frac{4}{5}\) and \(270^\circ < \theta < 360^\circ\), follow these steps:
1. Determine \(\cos \theta\):
We are given that \(\sin \theta = -\frac{4}{5}\). Since \(\sin^2 \theta + \cos^2 \theta = 1\), we can solve for \(\cos \theta\):
[tex]\[ \sin^2 \theta + \cos^2 \theta = 1 \][/tex]
[tex]\[ \left(-\frac{4}{5}\right)^2 + \cos^2 \theta = 1 \][/tex]
[tex]\[ \frac{16}{25} + \cos^2 \theta = 1 \][/tex]
[tex]\[ \cos^2 \theta = 1 - \frac{16}{25} \][/tex]
[tex]\[ \cos^2 \theta = \frac{25}{25} - \frac{16}{25} \][/tex]
[tex]\[ \cos^2 \theta = \frac{9}{25} \][/tex]
Thus, \(\cos \theta = \pm \frac{3}{5}\).
Since \(\theta\) is between \(270^\circ\) and \(360^\circ\) (which is the fourth quadrant), \(\cos \theta\) is positive in this range:
[tex]\[ \cos \theta = \frac{3}{5} \][/tex]
2. Calculate \(\sin 2\theta\):
Using the double-angle formula for sine:
[tex]\[ \sin 2\theta = 2 \sin \theta \cos \theta \][/tex]
[tex]\[ \sin 2\theta = 2 \left(-\frac{4}{5}\right) \left(\frac{3}{5}\right) \][/tex]
[tex]\[ \sin 2\theta = 2 \left(-\frac{12}{25}\right) \][/tex]
[tex]\[ \sin 2\theta = -\frac{24}{25} \][/tex]
By approximating, this value is \(-0.96\).
3. Calculate \(\cos 2\theta\):
Using the double-angle formula for cosine:
[tex]\[ \cos 2\theta = \cos^2 \theta - \sin^2 \theta \][/tex]
[tex]\[ \cos 2\theta = \left(\frac{3}{5}\right)^2 - \left(-\frac{4}{5}\right)^2 \][/tex]
[tex]\[ \cos 2\theta = \frac{9}{25} - \frac{16}{25} \][/tex]
[tex]\[ \cos 2\theta = \frac{9 - 16}{25} \][/tex]
[tex]\[ \cos 2\theta = -\frac{7}{25} \][/tex]
By approximating, this value is \(-0.28\).
4. Calculate \(\tan 2\theta\):
Using the relationship between sine, cosine, and tangent:
[tex]\[ \tan 2\theta = \frac{\sin 2\theta}{\cos 2\theta} \][/tex]
[tex]\[ \tan 2\theta = \frac{-\frac{24}{25}}{-\frac{7}{25}} \][/tex]
[tex]\[ \tan 2\theta = \frac{24}{7} \][/tex]
By approximating, this value is approximately \(3.43\).
Therefore, the exact values are:
[tex]\[ \sin 2\theta = -\frac{24}{25} \approx -0.96 \][/tex]
[tex]\[ \cos 2\theta = -\frac{7}{25} \approx -0.28 \][/tex]
[tex]\[ \tan 2\theta = \frac{24}{7} \approx 3.43 \][/tex]
These are the exact trigonometric values for the given angle [tex]\(\theta\)[/tex].
1. Determine \(\cos \theta\):
We are given that \(\sin \theta = -\frac{4}{5}\). Since \(\sin^2 \theta + \cos^2 \theta = 1\), we can solve for \(\cos \theta\):
[tex]\[ \sin^2 \theta + \cos^2 \theta = 1 \][/tex]
[tex]\[ \left(-\frac{4}{5}\right)^2 + \cos^2 \theta = 1 \][/tex]
[tex]\[ \frac{16}{25} + \cos^2 \theta = 1 \][/tex]
[tex]\[ \cos^2 \theta = 1 - \frac{16}{25} \][/tex]
[tex]\[ \cos^2 \theta = \frac{25}{25} - \frac{16}{25} \][/tex]
[tex]\[ \cos^2 \theta = \frac{9}{25} \][/tex]
Thus, \(\cos \theta = \pm \frac{3}{5}\).
Since \(\theta\) is between \(270^\circ\) and \(360^\circ\) (which is the fourth quadrant), \(\cos \theta\) is positive in this range:
[tex]\[ \cos \theta = \frac{3}{5} \][/tex]
2. Calculate \(\sin 2\theta\):
Using the double-angle formula for sine:
[tex]\[ \sin 2\theta = 2 \sin \theta \cos \theta \][/tex]
[tex]\[ \sin 2\theta = 2 \left(-\frac{4}{5}\right) \left(\frac{3}{5}\right) \][/tex]
[tex]\[ \sin 2\theta = 2 \left(-\frac{12}{25}\right) \][/tex]
[tex]\[ \sin 2\theta = -\frac{24}{25} \][/tex]
By approximating, this value is \(-0.96\).
3. Calculate \(\cos 2\theta\):
Using the double-angle formula for cosine:
[tex]\[ \cos 2\theta = \cos^2 \theta - \sin^2 \theta \][/tex]
[tex]\[ \cos 2\theta = \left(\frac{3}{5}\right)^2 - \left(-\frac{4}{5}\right)^2 \][/tex]
[tex]\[ \cos 2\theta = \frac{9}{25} - \frac{16}{25} \][/tex]
[tex]\[ \cos 2\theta = \frac{9 - 16}{25} \][/tex]
[tex]\[ \cos 2\theta = -\frac{7}{25} \][/tex]
By approximating, this value is \(-0.28\).
4. Calculate \(\tan 2\theta\):
Using the relationship between sine, cosine, and tangent:
[tex]\[ \tan 2\theta = \frac{\sin 2\theta}{\cos 2\theta} \][/tex]
[tex]\[ \tan 2\theta = \frac{-\frac{24}{25}}{-\frac{7}{25}} \][/tex]
[tex]\[ \tan 2\theta = \frac{24}{7} \][/tex]
By approximating, this value is approximately \(3.43\).
Therefore, the exact values are:
[tex]\[ \sin 2\theta = -\frac{24}{25} \approx -0.96 \][/tex]
[tex]\[ \cos 2\theta = -\frac{7}{25} \approx -0.28 \][/tex]
[tex]\[ \tan 2\theta = \frac{24}{7} \approx 3.43 \][/tex]
These are the exact trigonometric values for the given angle [tex]\(\theta\)[/tex].
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