1. Let's start with the basics of trigonometry: Trigonometry studies 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 an angle $\theta$ in a right triangle: - $\sin\theta = \frac{\text{opposite side}}{\text{hypotenuse}}$ - $\cos\theta = \frac{\text{adjacent side}}{\text{hypotenuse}}$ - $\tan\theta = \frac{\text{opposite side}}{\text{adjacent side}}$ 3. The reciprocal functions are cosecant ($\csc$), secant ($\sec$), and cotangent ($\cot$): - $\csc\theta = \frac{1}{\sin\theta}$ - $\sec\theta = \frac{1}{\cos\theta}$ - $\cot\theta = \frac{1}{\tan\theta}$ 4. Key identities include the Pythagorean identity: $$\sin^2\theta + \cos^2\theta = 1$$ 5. Additional angle formulas include the sum and difference identities, e.g.: - $\sin(a\pm b) = \sin a \cos b \pm \cos a \sin b$ - $\cos(a\pm b) = \cos a \cos b \mp \sin a \sin b$ 6. Trigonometry is applied to solve problems involving angles and distances, analyze waves, oscillations, and circular motion, among other physical phenomena. This guide covers fundamentals essential for understanding and applying trigonometry.