(c) \( 2 \tan 2 \theta=-\sqrt{5} \) with \( 0^{\circ}<2 \theta<180^{\circ} \), determine the value of the following without the tse of a calculator and with the aid of a diagram: \( \begin{array}{lll}\text { (1) } \sin 2 \theta & \text { (2) } \sin \theta \cos \theta & \text { (3) } 2 \cos ^{2} \theta \\ \text { (4) } \cos \theta & \end{array} \)

To solve the equation \( 2 \tan 2\theta = -\sqrt{5} \) with the condition \( 0^{\circ} < 2\theta < 180^{\circ} \), we will follow these steps: ### Step 1: Solve for \( \tan 2\theta \) First, we can simplify the equation: \[ \tan 2\theta = -\frac{\sqrt{5}}{2} \] ### Step 2: Determine the angle \( 2\theta \) The tangent function is negative in the second quadrant. Therefore, we can find the reference angle \( \alpha \) such that: \[ \tan \alpha = \frac{\sqrt{5}}{2} \] Using the inverse tangent function, we can find \( \alpha \): \[ \alpha = \tan^{-1}\left(\frac{\sqrt{5}}{2}\right) \] Since \( 2\theta \) is in the second quadrant, we have: \[ 2\theta = 180^{\circ} - \alpha \] ### Step 3: Calculate \( \sin 2\theta \) Using the identity \( \sin 2\theta = \frac{2\tan 2\theta}{1 + \tan^2 2\theta} \): 1. Calculate \( \tan^2 2\theta \): \[ \tan^2 2\theta = \left(-\frac{\sqrt{5}}{2}\right)^2 = \frac{5}{4} \] 2. Substitute into the sine formula: \[ \sin 2\theta = \frac{2\left(-\frac{\sqrt{5}}{2}\right)}{1 + \frac{5}{4}} = \frac{-\sqrt{5}}{\frac{9}{4}} = -\frac{4\sqrt{5}}{9} \] ### Step 4: Calculate \( \sin \theta \cos \theta \) Using the identity \( \sin 2\theta = 2 \sin \theta \cos \theta \): \[ 2 \sin \theta \cos \theta = -\frac{4\sqrt{5}}{9} \] Thus, \[ \sin \theta \cos \theta = -\frac{2\sqrt{5}}{9} \] ### Step 5: Calculate \( 2 \cos^2 \theta \) Using the identity \( \cos^2 \theta = \frac{1 + \cos 2\theta}{2} \): 1. First, we need to find \( \cos 2\theta \): Using the identity \( \cos 2\theta = \frac{1 - \tan^2 2\theta}{1 + \tan^2 2\theta} \): \[ \cos 2\theta = \frac{1 - \frac{5}{4}}{1 + \frac{5}{4}} = \frac{-\frac{1}{4}}{\frac{9}{4}} = -\frac{1}{9} \] 2. Now substitute into the cosine formula: \[ 2 \cos^2 \theta = 1 + \cos 2\theta = 1 - \frac{1}{9} = \frac{8}{9} \] ### Step 6: Calculate \( \cos \theta \) Using the identity \( \cos^2 \theta = \frac{1 + \cos 2\theta}{2} \): \[ \cos^2 \theta = \frac{1 - \frac{1}{9}}{2} = \frac{\frac{8}{9}}{2} = \frac{4}{9} \] Thus, \[ \cos \theta = \sqrt{\frac{4}{9}} = \frac{2}{3} \] ### Summary of Results 1. \( \sin 2\theta = -\frac{4\sqrt{5}}{9} \) 2. \( \sin \theta \cos \theta = -\frac{2\sqrt{5}}{9} \) 3. \( 2 \cos^2 \theta = \frac{8}{9} \) 4. \( \cos \theta = \frac{2}{3} \) These are the values for the given expressions.