If \( 2 \sin \theta+7=0 \) and \( 90^{\circ}<\theta<270^{\circ} \), Determine the value of \( 24 \tan \theta+25 \cos \theta \) with the aid of a diagram. If \( \cos 20^{\circ}=p \), express each of the following in terms of \( p \) : \( 5.2 .1 \cos \left(-20^{\circ}\right) \) \( 5.2 .2 \cos 160^{\circ} \)

Let's break down the problem step by step. ### Step 1: Solve the equation \( 2 \sin \theta + 7 = 0 \) First, we can isolate \( \sin \theta \): \[ 2 \sin \theta = -7 \] \[ \sin \theta = -\frac{7}{2} \] However, since the sine function has a range of \([-1, 1]\), there are no valid solutions for \( \sin \theta = -\frac{7}{2} \). Therefore, we need to check if there was a misunderstanding in the problem statement. Assuming the equation should be \( 2 \sin \theta - 7 = 0 \): \[ 2 \sin \theta = 7 \] \[ \sin \theta = \frac{7}{2} \] This is also invalid. Let's assume the equation is correct and proceed with the next part of the problem. ### Step 2: Determine \( 24 \tan \theta + 25 \cos \theta \) Since we cannot find \( \theta \) from the sine equation, we will assume \( \theta \) is in the third quadrant (as given \( 90^{\circ} < \theta < 270^{\circ} \)) and proceed to find \( 24 \tan \theta + 25 \cos \theta \) using the known properties of trigonometric functions. ### Step 3: Express \( 5.2.1 \cos(-20^{\circ}) \) and \( 5.2.2 \cos(160^{\circ}) \) in terms of \( p \) Given \( \cos 20^{\circ} = p \): 1. **For \( 5.2.1 \cos(-20^{\circ}) \)**: \[ \cos(-20^{\circ}) = \cos(20^{\circ}) = p \] Thus, \[ 5.2.1 = 5.2 \cdot 1 \cdot p = 5.2p \] 2. **For \( 5.2.2 \cos(160^{\circ}) \)**: Using the cosine identity: \[ \cos(160^{\circ}) = \cos(180^{\circ} - 20^{\circ}) = -\cos(20^{\circ}) = -p \] Thus, \[ 5.2.2 = 5.2 \cdot (-p) = -5.2p \] ### Final Results 1. Since \( \sin \theta = -\frac{7}{2} \) is invalid, we cannot determine \( 24 \tan \theta + 25 \cos \theta \) without a valid \( \theta \). 2. The expressions in terms of \( p \) are: - \( 5.2.1 = 5.2p \) - \( 5.2.2 = -5.2p \) If you have any further clarifications or corrections regarding the sine equation, please let me know!