1. Let's start by understanding what trigonometric ratios are. 2. Trigonometric ratios are relationships between the lengths of the sides of a right triangle. 3. Consider a right triangle with an angle $\theta$, the side opposite to $\theta$, the side adjacent to $\theta$, and the hypotenuse (the longest side). 4. The primary trigonometric ratios are: - Sine ($\sin \theta$): ratio of the length of the opposite side to the hypotenuse. $$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$$ - Cosine ($\cos \theta$): ratio of the length of the adjacent side to the hypotenuse. $$\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$$ - Tangent ($\tan \theta$): ratio of the length of the opposite side to the adjacent side. $$\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$$ 5. There are also three reciprocal ratios: - Cosecant ($\csc \theta$): reciprocal of sine. $$\csc \theta = \frac{1}{\sin \theta} = \frac{\text{hypotenuse}}{\text{opposite}}$$ - Secant ($\sec \theta$): reciprocal of cosine. $$\sec \theta = \frac{1}{\cos \theta} = \frac{\text{hypotenuse}}{\text{adjacent}}$$ - Cotangent ($\cot \theta$): reciprocal of tangent. $$\cot \theta = \frac{1}{\tan \theta} = \frac{\text{adjacent}}{\text{opposite}}$$ 6. These ratios help us solve problems involving right triangles and periodic functions.