Question 2 (a) - Sketch of the Function $T(t) = 75 - 45e^{-0.2t}$

Graph of $T(t) = 75 - 45e^{-0.2t}$

To sketch the function, we identify the following points:

• At $t = 0$: $T(0) = 75 - 45e^{-0.2 \times 0} = 75 - 45 = 30$
• As $t \to \infty$: $T(t) \to 75$

The horizontal asymptote is $T = 75$.

\hline t & T(t) \\ \hline 0 & 30 \\ 10 & 75 - 45e^{-2} \approx 66.94 \\ 20 & 75 - 45e^{-4} \approx 74.18 \\ 30 & 75 - 45e^{-6} \approx 74.94 \\ \hline \end{array}$$ Using these points, we can sketch the function. ### Sketch of the Function $$\begin{array}{c} \includegraphics[scale=0.5]{/mnt/data/sketch_t_function.png} \end{array}$$ ### Question 3 - Sketches and Graphs of the Angles #### (a) $$90°$$ The angle $$90°$$ is a right angle on the positive y-axis. #### Sketch: ``` | | | | | |________ 90° ``` #### (b) $$-150°$$ The angle $$-150°$$ is in the third quadrant, and it's $$150°$$ measured clockwise from the positive x-axis. #### Sketch: ``` \ \ \ \ \________ -150° ``` #### (c) $$\frac{3\pi}{4}$$ The angle $$\frac{3\pi}{4}$$ (or $$135°$$) is in the second quadrant, and it's $$45°$$ counterclockwise from the negative x-axis. #### Sketch: ``` / / / /________ 135° ``` #### (d) $$-\frac{5\pi}{3}$$ The angle $$-\frac{5\pi}{3}$$ (or $$-300°$$) is in the first quadrant, and it's $$60°$$ measured clockwise from the positive x-axis. #### Sketch: ``` | | |_________ -300° ``` ### Graphs of the Trigonometric Functions #### (a) $$90°$$ $$\sin(90°) = 1$$ $$\cos(90°) = 0$$ $$\tan(90°) = \text{undefined}$$ #### (b) $$-150°$$ $$\sin(-150°) = -\sin(150°) = -\frac{1}{2}$$ $$\cos(-150°) = \cos(150°) = -\frac{\sqrt{3}}{2}$$ $$\tan(-150°) = \frac{\sin(-150°)}{\cos(-150°)} = \frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = -\frac{1}{\sqrt{3}}$$ #### (c) $$\frac{3\pi}{4}$$ $$\sin\left(\frac{3\pi}{4}\right) = \sin\left(180° - 45°\right) = \sin(45°) = \frac{\sqrt{2}}{2}$$ $$\cos\left(\frac{3\pi}{4}\right) = -\cos(45°) = -\frac{\sqrt{2}}{2}$$ $$\tan\left(\frac{3\pi}{4}\right) = \frac{\sin\left(\frac{3\pi}{4}\right)}{\cos\left(\frac{3\pi}{4}\right)} = -1$$ #### (d) $$-\frac{5\pi}{3}$$ $$\sin\left(-\frac{5\pi}{3}\right) = -\sin\left(\frac{5\pi}{3}\right) = -\sin\left(300°\right) = -\left(-\frac{\sqrt{3}}{2}\right) = \frac{\sqrt{3}}{2}$$ $$\cos\left(-\frac{5\pi}{3}\right) = \cos\left(\frac{5\pi}{3}\right) = \cos\left(300°\right) = \frac{1}{2}$$ $$\tan\left(-\frac{5\pi}{3}\right) = \frac{\sin\left(-\frac{5\pi}{3}\right)}{\cos\left(-\frac{5\pi}{3}\right)} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3}$$ --- Would you like more details on any of the solutions, or do you have any further questions? Here are some relative questions to consider: 1. How do you solve logarithmic equations in general? 2. What is the general form of an exponential decay function? 3. How can we determine the horizontal asymptote of a given function? 4. What are the properties of sine, cosine, and tangent functions? 5. How do you convert degrees to radians and vice versa? 6. How do you find the inverse of an exponential function? 7. What is the significance of the unit circle in trigonometry? 8. How do you graph trigonometric functions? **Tip:** Always double-check your work for arithmetic or algebraic mistakes to ensure accuracy.