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].
