1. **Understanding the basic trigonometric functions:** The primary functions are sine ($\sin x$), cosine ($\cos x$), and tangent ($\tan x$). Each has a characteristic wave shape and period. 2. **Key properties of sine and cosine graphs:** - Amplitude: The height from the center line to the peak, given by $|A|$ in $y = A\sin(Bx + C) + D$ or $y = A\cos(Bx + C) + D$. - Period: The length of one complete cycle, calculated by $\frac{2\pi}{|B|}$. - Phase shift: Horizontal shift, given by $-\frac{C}{B}$. - Vertical shift: Up or down movement, given by $D$. 3. **Tangent graph properties:** - Period: $\frac{\pi}{|B|}$. - Vertical asymptotes occur where $\cos(Bx + C) = 0$. - Amplitude is not defined as tangent values can be very large. 4. **Graph transformations:** - Vertical stretch/compression by $A$. - Horizontal stretch/compression by $\frac{1}{B}$. - Horizontal shift by phase shift. - Vertical shift by $D$. 5. **Solving trigonometric equations:** - Isolate the trig function. - Use inverse trig functions to find principal solutions. - Use periodicity to find general solutions, e.g., for sine and cosine: $x = \theta + 2n\pi$ or $x = \pi - \theta + 2n\pi$, for integer $n$. - For tangent: $x = \theta + n\pi$. 6. **Example:** Solve $2\sin(3x - \frac{\pi}{4}) = 1$. - Step 1: Isolate sine: $\sin(3x - \frac{\pi}{4}) = \frac{1}{2}$. - Step 2: Find principal solutions: $3x - \frac{\pi}{4} = \frac{\pi}{6}$ or $3x - \frac{\pi}{4} = \pi - \frac{\pi}{6} = \frac{5\pi}{6}$. - Step 3: Solve for $x$: $$3x = \frac{\pi}{6} + \frac{\pi}{4} = \frac{2\pi}{12} + \frac{3\pi}{12} = \frac{5\pi}{12}$$ $$x = \frac{5\pi}{36} + \frac{2n\pi}{3}$$ and $$3x = \frac{5\pi}{6} + \frac{\pi}{4} = \frac{10\pi}{12} + \frac{3\pi}{12} = \frac{13\pi}{12}$$ $$x = \frac{13\pi}{36} + \frac{2n\pi}{3}$$ where $n$ is any integer. 7. **Summary:** Know the shape and properties of trig graphs, how to apply transformations, and how to solve equations using inverse functions and periodicity.