Let's work through the questions one by one.

4a. Find the period of $f(x) = -2 \tan(3x)$:

The period of a tangent function is determined by the formula: $\text{Period} = \frac{\pi}{|b|}$ where $b$ is the coefficient of $x$ in $\tan(bx)$.

Here, $b = 3$, so: $\text{Period} = \frac{\pi}{3}$

4b. Graph one cycle of $f(x) = -2 \tan(3x)$:

1. Key points of $\tan(3x)$:

   • The tangent function has vertical asymptotes where $3x = \pm\frac{\pi}{2}$, or $x = \pm\frac{\pi}{6}$ for one period of $\tan(3x)$.
   • The midline is at $y = 0$, and the function passes through the origin at $(0, 0)$.

2. Amplitude: The scaling factor of -2 reflects the graph vertically and multiplies the height of each point by 2.

3. Cycle:

   • Asymptotes: $x = -\frac{\pi}{6}$ and $x = \frac{\pi}{6}$
   • Points:
     • At $x = 0$, $f(0) = 0$
     • At $x = \pm \frac{\pi}{12}$, $f\left(\pm\frac{\pi}{12}\right) = \mp 2$

Graph these points and the asymptotes to plot one period.

5. Simplify $\cos(x - \frac{\pi}{3}) + \sin(x + \frac{\pi}{6})$:

Using the angle addition/subtraction formulas:

1. Expand: $\cos(x - \frac{\pi}{3}) = \cos x \cos \frac{\pi}{3} + \sin x \sin \frac{\pi}{3}$ $\sin(x + \frac{\pi}{6}) = \sin x \cos \frac{\pi}{6} + \cos x \sin \frac{\pi}{6}$

2. Substitute values for trigonometric constants:

   • $\cos \frac{\pi}{3} = \frac{1}{2}, \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}$
   • $\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}, \sin \frac{\pi}{6} = \frac{1}{2}$

   Substituting, we get: $\cos(x - \frac{\pi}{3}) = \frac{1}{2}\cos x + \frac{\sqrt{3}}{2}\sin x$ $\sin(x + \frac{\pi}{6}) = \frac{\sqrt{3}}{2}\sin x + \frac{1}{2}\cos x$

3. Combine terms: $\cos(x - \frac{\pi}{3}) + \sin(x + \frac{\pi}{6}) = \left(\frac{1}{2} + \frac{1}{2}\right)\cos x + \left(\frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2}\right)\sin x$

4. Simplify: $\cos(x - \frac{\pi}{3}) + \sin(x + \frac{\pi}{6}) = \cos x + \sqrt{3}\sin x$

Final Solutions:

• 4a: Period = $\frac{\pi}{3}$
• 4b: Graph one cycle from $x = -\frac{\pi}{6}$ to $x = \frac{\pi}{6}$, with points $(0, 0)$, $(\pm\frac{\pi}{12}, \mp2)$, and asymptotes at $x = \pm\frac{\pi}{6}$.
• 5: Simplified expression = $\cos x + \sqrt{3}\sin x$.

Would you like me to graph the tangent function or explain any steps in further detail?

Questions for Further Learning:

1. What are the general properties of tangent functions with transformations?
2. How do vertical reflections affect the graph of trigonometric functions?
3. Why does the period of a tangent function differ from sine and cosine?
4. How can you verify the simplified trigonometric expression using a calculator?
5. Can you derive the general period formula for transformed tangent functions?

Tip:

For trigonometric graphs, always start by marking asymptotes and key points to avoid errors in the cycle.