Page 1: Understanding Trigonometric Ratios and Slopes
This page provides a comprehensive overview of trigonometric ratios and their practical applications. The content begins with the fundamental definitions of sine, cosine, and tangent in relation to right triangles.
Definition: Sine (sin) is the ratio of the opposite side to the hypotenuse, cosine (cos) is the ratio of the adjacent side to the hypotenuse, and tangent (tan) is the ratio of the opposite side to the adjacent side.
Example: A right triangle with sides 3cm, 4cm, and 5cm is used to demonstrate:
- sin(θ) = 3/5 = 0.6
- cos(θ) = 4/5 = 0.8
- tan(θ) = 3/4 = 0.75
Highlight: The relationship between trigonometric functions is demonstrated through the fundamental identity: sin²(θ) + cos²(θ) = 1
Vocabulary:
- Gegenkathete: opposite side
- Ankathete: adjacent side
- Hypotenuse: longest side of a right triangle
Example: A practical slope calculation is presented with a 6% grade over 100m:
- tan(θ) = 6m/100m = 0.06
- θ = arctan(0.06) ≈ 3.34°
The page concludes with important reference values for common angles and demonstrates how these concepts apply to real-world slope calculations in engineering and construction contexts.
