1. Trigonometry is the branch of mathematics that studies the relationships between the angles and sides of triangles. 2. The primary functions in trigonometry are sine ($\sin$), cosine ($\cos$), and tangent ($\tan$), which relate the angles of a right triangle to the ratios of its sides. 3. For a right triangle with an angle $\theta$, the definitions are: - $\sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}}$ - $\cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}}$ - $\tan \theta = \frac{\text{opposite side}}{\text{adjacent side}}$ 4. Important rules include the Pythagorean identity: $$\sin^2 \theta + \cos^2 \theta = 1$$ 5. Trigonometry is used to solve for unknown sides or angles in triangles, model periodic phenomena, and in many fields such as physics, engineering, and navigation. 6. Other functions include cosecant ($\csc$), secant ($\sec$), and cotangent ($\cot$), which are reciprocals of sine, cosine, and tangent respectively. 7. The unit circle is a fundamental tool in trigonometry, where angles correspond to points on a circle of radius 1, linking trigonometric functions to coordinates. 8. Trigonometric equations can be solved using identities, inverse functions, and algebraic manipulation. 9. Overall, trigonometry provides a powerful framework to analyze and understand angles and distances in various contexts.