4. The angle of depression from the top of a building to a point on the ground is \( 32^{\circ} 30^{\circ} \). How far is the point on the ground from the top of the butlivgs 1 fine brating in 28 high? 5. Find all values of \( \theta \), if \( \theta \) is in the interval \( \left(0^{\circ}, 360^{\circ}\right) \) and has the following function value:

To solve the problems you've presented, let's break them down one by one. ### Problem 4: Angle of Depression The angle of depression from the top of a building to a point on the ground is given as \( 32^\circ 30' \). First, we need to convert this angle into decimal degrees: \[ 32^\circ 30' = 32 + \frac{30}{60} = 32.5^\circ \] Now, we can use the angle of depression to find the horizontal distance from the base of the building to the point on the ground. The height of the building is given as 28 units (assuming meters, feet, etc.). Using the tangent function, we have: \[ \tan(\text{angle of depression}) = \frac{\text{height of the building}}{\text{horizontal distance}} \] Let \( d \) be the horizontal distance. Then: \[ \tan(32.5^\circ) = \frac{28}{d} \] Rearranging this gives: \[ d = \frac{28}{\tan(32.5^\circ)} \] Now, we can calculate \( \tan(32.5^\circ) \): \[ \tan(32.5^\circ) \approx 0.6349 \quad (\text{using a calculator}) \] Now substituting this value back into the equation for \( d \): \[ d = \frac{28}{0.6349} \approx 44.1 \] Thus, the horizontal distance from the point on the ground to the base of the building is approximately **44.1 units**. ### Problem 5: Finding Values of \( \theta \) You mentioned finding all values of \( \theta \) in the interval \( (0^\circ, 360^\circ) \) for a certain function value, but you did not specify the function. Common trigonometric functions include sine, cosine, and tangent. If you provide the specific function value (e.g., \( \sin(\theta) = k \), \( \cos(\theta) = k \), or \( \tan(\theta) = k \)), I can help you find the values of \( \theta \) accordingly. Please provide that information!