1. Find 6 Tig ratios using the b) $$ \sin heta= $$ b) $$ \cos heta= $$ c) $$ an heta= $$ e) $$ \cot heta= $$ e) $$ \sec heta= $$ f) $$ \sin heta= $$ $$ \csc heta= $$

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1. Find 6 Tig ratios using the b) $$ \sin 	heta= $$ b) $$ \cos 	heta= $$ c) $$ 	an 	heta= $$ e) $$ \cot 	heta= $$ e) $$ \sec 	heta= $$ f) $$ \sin 	heta= $$ $$ \csc 	heta= $$

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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!
Here are the six trigonometric ratios for angle $$ \theta $$ in a right triangle: 1. **Sine ($$ \sin \theta $$)**: $$ \frac{\text{Opposite}}{\text{Hypotenuse}} $$ 2. **Cosine ($$ \cos \theta $$)**: $$ \frac{\text{Adjacent}}{\text{Hypotenuse}} $$ 3. **Tangent ($$ \tan \theta $$)**: $$ \frac{\text{Opposite}}{\text{Adjacent}} $$ 4. **Cotangent ($$ \cot \theta $$)**: $$ \frac{\text{Adjacent}}{\text{Opposite}} $$ 5. **Secant ($$ \sec \theta $$)**: $$ \frac{\text{Hypotenuse}}{\text{Adjacent}} $$ 6. **Cosecant ($$ \csc \theta $$)**: $$ \frac{\text{Hypotenuse}}{\text{Opposite}} $$ These ratios help determine the relationships between the sides of a right triangle relative to angle $$ \theta $$.

Find 6 Tig ratios using the
b)
b)
c)
e)
e)
f)

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