Answer :
To find the exact values of \(\sin 2\theta\), \(\cos 2\theta\), and \(\tan 2\theta\) for \(\theta\) in the interval \(90^\circ < \theta < 180^\circ\) given that \(\sec \theta = -\frac{3}{2}\), we can proceed with the following steps:
### Step 1: Find \(\cos \theta\)
Since \(\sec \theta = \frac{1}{\cos \theta}\):
[tex]\[ \sec \theta = -\frac{3}{2} \implies \cos \theta = -\frac{2}{3} \][/tex]
### Step 2: Determine \(\sin \theta\)
We use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\):
[tex]\[ \sin^2 \theta + \left(-\frac{2}{3}\right)^2 = 1 \][/tex]
[tex]\[ \sin^2 \theta + \frac{4}{9} = 1 \][/tex]
[tex]\[ \sin^2 \theta = 1 - \frac{4}{9} \][/tex]
[tex]\[ \sin^2 \theta = \frac{9}{9} - \frac{4}{9} \][/tex]
[tex]\[ \sin^2 \theta = \frac{5}{9} \][/tex]
[tex]\[ \sin \theta = \pm \sqrt{\frac{5}{9}} \][/tex]
Since \(90^\circ < \theta < 180^\circ\) and sine is positive in this interval, we have:
[tex]\[ \sin \theta = \sqrt{\frac{5}{9}} = \frac{\sqrt{5}}{3} \][/tex]
### Step 3: Find \(\sin 2\theta\)
Using the double angle formula for sine:
[tex]\[ \sin 2\theta = 2 \sin \theta \cos \theta \][/tex]
Substitute \(\sin \theta = \frac{\sqrt{5}}{3}\) and \(\cos \theta = -\frac{2}{3}\):
[tex]\[ \sin 2\theta = 2 \left(\frac{\sqrt{5}}{3}\right) \left(-\frac{2}{3}\right) \][/tex]
[tex]\[ \sin 2\theta = 2 \times \frac{\sqrt{5}}{3} \times -\frac{2}{3} \][/tex]
[tex]\[ \sin 2\theta = \frac{2\sqrt{5} \times -2}{9} \][/tex]
[tex]\[ \sin 2\theta = -\frac{4\sqrt{5}}{9} \][/tex]
Numerically, this is approximately:
[tex]\[ \sin 2\theta \approx -0.9938079899999065 \][/tex]
### Step 4: Find \(\cos 2\theta\)
Using the double angle formula for cosine:
[tex]\[ \cos 2\theta = \cos^2 \theta - \sin^2 \theta \][/tex]
Substitute \(\cos \theta = -\frac{2}{3}\) and \(\sin \theta = \frac{\sqrt{5}}{3}\):
[tex]\[ \cos 2\theta = \left(-\frac{2}{3}\right)^2 - \left(\frac{\sqrt{5}}{3}\right)^2 \][/tex]
[tex]\[ \cos 2\theta = \frac{4}{9} - \frac{5}{9} \][/tex]
[tex]\[ \cos 2\theta = -\frac{1}{9} \][/tex]
Numerically, this is approximately:
[tex]\[ \cos 2\theta \approx -0.11111111111111116 \][/tex]
### Step 5: Find \(\tan 2\theta\)
Using the relationship \(\tan 2\theta = \frac{\sin 2\theta}{\cos 2\theta}\):
[tex]\[ \tan 2\theta = \frac{-\frac{4\sqrt{5}}{9}}{-\frac{1}{9}} \][/tex]
[tex]\[ \tan 2\theta = \frac{4\sqrt{5}}{1} \][/tex]
[tex]\[ \tan 2\theta = 4\sqrt{5} \][/tex]
Numerically, this is approximately:
[tex]\[ \tan 2\theta \approx 8.944271909999154 \][/tex]
Thus, the exact values are:
[tex]\[ \sin 2\theta = -\frac{4\sqrt{5}}{9}, \quad \cos 2\theta = -\frac{1}{9}, \quad \tan 2\theta = 4\sqrt{5} \][/tex]
The numerical approximations are:
[tex]\[ \sin 2\theta \approx -0.9938079899999065, \quad \cos 2\theta \approx -0.11111111111111116, \quad \tan 2\theta \approx 8.944271909999154 \][/tex]
### Step 1: Find \(\cos \theta\)
Since \(\sec \theta = \frac{1}{\cos \theta}\):
[tex]\[ \sec \theta = -\frac{3}{2} \implies \cos \theta = -\frac{2}{3} \][/tex]
### Step 2: Determine \(\sin \theta\)
We use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\):
[tex]\[ \sin^2 \theta + \left(-\frac{2}{3}\right)^2 = 1 \][/tex]
[tex]\[ \sin^2 \theta + \frac{4}{9} = 1 \][/tex]
[tex]\[ \sin^2 \theta = 1 - \frac{4}{9} \][/tex]
[tex]\[ \sin^2 \theta = \frac{9}{9} - \frac{4}{9} \][/tex]
[tex]\[ \sin^2 \theta = \frac{5}{9} \][/tex]
[tex]\[ \sin \theta = \pm \sqrt{\frac{5}{9}} \][/tex]
Since \(90^\circ < \theta < 180^\circ\) and sine is positive in this interval, we have:
[tex]\[ \sin \theta = \sqrt{\frac{5}{9}} = \frac{\sqrt{5}}{3} \][/tex]
### Step 3: Find \(\sin 2\theta\)
Using the double angle formula for sine:
[tex]\[ \sin 2\theta = 2 \sin \theta \cos \theta \][/tex]
Substitute \(\sin \theta = \frac{\sqrt{5}}{3}\) and \(\cos \theta = -\frac{2}{3}\):
[tex]\[ \sin 2\theta = 2 \left(\frac{\sqrt{5}}{3}\right) \left(-\frac{2}{3}\right) \][/tex]
[tex]\[ \sin 2\theta = 2 \times \frac{\sqrt{5}}{3} \times -\frac{2}{3} \][/tex]
[tex]\[ \sin 2\theta = \frac{2\sqrt{5} \times -2}{9} \][/tex]
[tex]\[ \sin 2\theta = -\frac{4\sqrt{5}}{9} \][/tex]
Numerically, this is approximately:
[tex]\[ \sin 2\theta \approx -0.9938079899999065 \][/tex]
### Step 4: Find \(\cos 2\theta\)
Using the double angle formula for cosine:
[tex]\[ \cos 2\theta = \cos^2 \theta - \sin^2 \theta \][/tex]
Substitute \(\cos \theta = -\frac{2}{3}\) and \(\sin \theta = \frac{\sqrt{5}}{3}\):
[tex]\[ \cos 2\theta = \left(-\frac{2}{3}\right)^2 - \left(\frac{\sqrt{5}}{3}\right)^2 \][/tex]
[tex]\[ \cos 2\theta = \frac{4}{9} - \frac{5}{9} \][/tex]
[tex]\[ \cos 2\theta = -\frac{1}{9} \][/tex]
Numerically, this is approximately:
[tex]\[ \cos 2\theta \approx -0.11111111111111116 \][/tex]
### Step 5: Find \(\tan 2\theta\)
Using the relationship \(\tan 2\theta = \frac{\sin 2\theta}{\cos 2\theta}\):
[tex]\[ \tan 2\theta = \frac{-\frac{4\sqrt{5}}{9}}{-\frac{1}{9}} \][/tex]
[tex]\[ \tan 2\theta = \frac{4\sqrt{5}}{1} \][/tex]
[tex]\[ \tan 2\theta = 4\sqrt{5} \][/tex]
Numerically, this is approximately:
[tex]\[ \tan 2\theta \approx 8.944271909999154 \][/tex]
Thus, the exact values are:
[tex]\[ \sin 2\theta = -\frac{4\sqrt{5}}{9}, \quad \cos 2\theta = -\frac{1}{9}, \quad \tan 2\theta = 4\sqrt{5} \][/tex]
The numerical approximations are:
[tex]\[ \sin 2\theta \approx -0.9938079899999065, \quad \cos 2\theta \approx -0.11111111111111116, \quad \tan 2\theta \approx 8.944271909999154 \][/tex]
