1. Let's start with the basics of trigonometry. Trigonometry deals with the relationships between the angles and sides of triangles, especially right triangles. 2. The primary functions are sine ($\sin$), cosine ($\cos$), and tangent ($\tan$). For a right triangle with angle $\theta$, opposite side $a$, adjacent side $b$, and hypotenuse $c$, the definitions are: $$\sin(\theta) = \frac{a}{c}, \quad \cos(\theta) = \frac{b}{c}, \quad \tan(\theta) = \frac{a}{b}$$ 3. Important rules include the Pythagorean identity: $$\sin^2(\theta) + \cos^2(\theta) = 1$$ 4. Let's solve an example: Find $\sin(\theta)$ if $\cos(\theta) = \frac{3}{5}$ and $\theta$ is in the first quadrant. 5. Using the Pythagorean identity: $$\sin^2(\theta) = 1 - \cos^2(\theta) = 1 - \left(\frac{3}{5}\right)^2 = 1 - \frac{9}{25} = \frac{16}{25}$$ 6. Taking the positive root (since $\theta$ is in the first quadrant where sine is positive): $$\sin(\theta) = \frac{4}{5}$$ 7. This shows how to find one trig function given another using identities and quadrant information. 8. We can also explore graphs of these functions, their periodicity, and applications in solving triangles.