1. Find 6 Tig ratios using the b) \( \sin \theta= \) b) \( \cos \theta= \) c) \( \tan \theta= \) e) \( \cot \theta= \) e) \( \sec \theta= \) f) \( \sin \theta= \) \( \csc \theta= \)

Certainly! To find the six trigonometric ratios for a given angle \( \theta \) in a right triangle, you'll need to understand the relationships between the sides of the triangle relative to \( \theta \). Here's a step-by-step guide: ### **Understanding the Right Triangle** Consider a right-angled triangle with the following components relative to angle \( \theta \): - **Opposite (O):** The side opposite to angle \( \theta \). - **Adjacent (A):** The side adjacent (next) to angle \( \theta \). - **Hypotenuse (H):** The longest side of the triangle, opposite the right angle. ### **The Six Trigonometric Ratios** 1. **Sine (\( \sin \theta \))** \[ \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{O}{H} \] **Interpretation:** It represents the ratio of the length of the side opposite \( \theta \) to the length of the hypotenuse. 2. **Cosine (\( \cos \theta \))** \[ \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{A}{H} \] **Interpretation:** It represents the ratio of the length of the side adjacent to \( \theta \) to the length of the hypotenuse. 3. **Tangent (\( \tan \theta \))** \[ \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{O}{A} \] **Interpretation:** It represents the ratio of the length of the side opposite \( \theta \) to the length of the side adjacent to \( \theta \). 4. **Cotangent (\( \cot \theta \))** \[ \cot \theta = \frac{\text{Adjacent}}{\text{Opposite}} = \frac{A}{O} \] **Interpretation:** It is the reciprocal of the tangent ratio, representing the ratio of the length of the side adjacent to \( \theta \) to the length of the side opposite \( \theta \). 5. **Secant (\( \sec \theta \))** \[ \sec \theta = \frac{\text{Hypotenuse}}{\text{Adjacent}} = \frac{H}{A} \] **Interpretation:** It is the reciprocal of the cosine ratio, representing the ratio of the hypotenuse to the side adjacent to \( \theta \). 6. **Cosecant (\( \csc \theta \))** \[ \csc \theta = \frac{\text{Hypotenuse}}{\text{Opposite}} = \frac{H}{O} \] **Interpretation:** It is the reciprocal of the sine ratio, representing the ratio of the hypotenuse to the side opposite \( \theta \). ### **Summary Table** | Trigonometric Function | Ratio Definition | |------------------------|----------------------------| | \( \sin \theta \) | \( \frac{O}{H} \) | | \( \cos \theta \) | \( \frac{A}{H} \) | | \( \tan \theta \) | \( \frac{O}{A} \) | | \( \cot \theta \) | \( \frac{A}{O} \) | | \( \sec \theta \) | \( \frac{H}{A} \) | | \( \csc \theta \) | \( \frac{H}{O} \) | ### **Example** Suppose you have a right triangle where: - Opposite side (\( O \)) = 3 units - Adjacent side (\( A \)) = 4 units - Hypotenuse (\( H \)) = 5 units Using the definitions: 1. \( \sin \theta = \frac{3}{5} = 0.6 \) 2. \( \cos \theta = \frac{4}{5} = 0.8 \) 3. \( \tan \theta = \frac{3}{4} = 0.75 \) 4. \( \cot \theta = \frac{4}{3} \approx 1.333 \) 5. \( \sec \theta = \frac{5}{4} = 1.25 \) 6. \( \csc \theta = \frac{5}{3} \approx 1.667 \) These ratios are fundamental in trigonometry and are widely used in various applications, including physics, engineering, and geometry. If you have specific values or a particular triangle you'd like to work with, feel free to provide more details, and I can help you calculate the exact ratios!