To find the other five trigonometric ratios (cosine, tangent, cosecant, secant, and cotangent) from the information given about angle 8, follow these steps:
1. **Understanding Sine**: The sine of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse. So, if sin(θ) = 12, it means:
- Opposite side = 12
- Hypotenuse = x (which we need to find).
2. **Using the Pythagorean Theorem**: Since we have the opposite side, we need to determine the adjacent side. Using the sine definition, we can find the hypotenuse:
To find the hypotenuse, we assume that the adjacent side is b. According to the Pythagorean theorem:
x² = 12² + b²But first, we need to find a ratio. Remember that sine ranges between -1 and 1 for angles in a triangle, and if 12 is a height, we may be using a different triangle definition. For a triangle with unit circle method, let’s denote:
sin(θ) = 12/rNow, rewrite:
r = 12/sin(θ)With the information that angle 8 is acute, we know:
3. **Finding Other Ratios**:
- Cosine (cos)**: Using cos(θ) = adjacent/hypotenuse.
- Tangent (tan)**: Remember tan(θ) = opposite/adjacent.
- Cosecant (csc)**: This is the reciprocal of sine: csc(θ) = 1/sin(θ).
- Secant (sec)**: The reciprocal of cosine: sec(θ) = 1/cos(θ).
- Cotangent (cot)**: The reciprocal of tangent: cot(θ) = 1/tan(θ).
4. **Calculation Example**: Suppose the hypotenuse r = 15:
- Cos(θ) = b/r
- Tan(θ) = 12/b
- Cosec(θ) = 15/12 = 1.25
- Sec(θ) = 15/b
- Cot(θ) = b/12
5. **Final Steps**: Calculate respective lengths and replace your values to get the ratios.
By completing these steps, you can derive the other five trigonometric ratios based on the initial sine value provided.