1. **Problem 1:** Evaluate $$F_3 = \frac{3}{2} a^3 + 2b - 5c + 5^x$$ given $$x=0$$ and $$y=\frac{\pi}{2}$$. Since $$x=0$$, we have $$5^x = 5^0 = 1$$. 2. **Problem 2:** Given $$y = \sin x$$, evaluate at $$y=\frac{\pi}{2}$$. Since $$y=\sin x$$, and $$y=\frac{\pi}{2}$$, this implies $$\sin x = \frac{\pi}{2}$$, which is not possible since $$\sin x$$ ranges between -1 and 1. So this is likely a statement of the function, not an evaluation. 3. **Problem 3:** Solve the system: $$4xy = 3 \sin(xy)$$ and $$F_3 = 5xy - 10 = 0$$ From the second equation, $$5xy - 10 = 0 \Rightarrow xy = 2$$. Substitute into the first: $$4 \times 2 = 3 \sin(2) \Rightarrow 8 = 3 \sin(2) \Rightarrow \sin(2) = \frac{8}{3}$$ But $$\sin(2)$$ cannot be greater than 1, so no real solution here. 4. **Problem 4:** From Table 1 and Table 2, compute: (1) Coordinates of new points $$x', y'$$ and $$x'', y''$$ by forward adjustment. (2) Bearing and distance of the line $$tr' - tr$$ after adjustment. **Step 1:** Convert bearings from degrees, minutes, seconds to decimal degrees. For example, bearing $$300^\circ 20' 45''$$: $$300 + \frac{20}{60} + \frac{45}{3600} = 300.3458^\circ$$ Similarly for $$82^\circ 14' 42''$$: $$82 + \frac{14}{60} + \frac{42}{3600} = 82.245^\circ$$ **Step 2:** Use the distances and bearings to compute coordinate changes: $$\Delta x = d \times \sin(\theta)$$ $$\Delta y = d \times \cos(\theta)$$ where $$d$$ is distance and $$\theta$$ is bearing in radians. **Step 3:** Calculate new coordinates by adding $$\Delta x$$ and $$\Delta y$$ to previous coordinates. **Step 4:** Calculate bearing and distance of line $$tr' - tr$$ using: $$\text{distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ $$\text{bearing} = \arctan\left(\frac{\Delta x}{\Delta y}\right)$$ adjusted to correct quadrant. 5. **Problem 5:** Define compass traversing terms: (i) True bearing: The angle measured clockwise from the north direction to the line, relative to true north. (ii) Magnetic bearing: The angle measured clockwise from magnetic north to the line. (iii) Forward bearing: The bearing of a line when moving from the starting point to the endpoint. (iv) Backward bearing: The bearing of a line when moving from the endpoint back to the starting point, differing by 180 degrees from the forward bearing. **Summary:** - Evaluated $$F_3$$ with given $$x$$. - Explained $$y=\sin x$$. - Analyzed system with $$xy$$. - Explained forward adjustment for coordinates. - Defined compass traversing terms.