Overview
This lecture introduces the basics of trigonometry in right triangles, covering the definitions and formulas for sine, cosine, and tangent, and how their values depend on the chosen angle.
Trigonometry in Right Triangles
- Trigonometry is based on right triangles and relates angles to side lengths.
- Each side is labeled typically as a, b, and c, where c is the hypotenuse (the side opposite the right angle).
Trigonometric Ratios and Formulas
- The value of trigonometric functions depends on the angle chosen as the reference.
- Sine (sin) of an angle = opposite side / hypotenuse.
- Cosine (cos) of an angle = adjacent side / hypotenuse.
- Tangent (tan) of an angle = opposite side / adjacent side.
- Use mnemonic devices to remember ratios (e.g., "SOH-CAH-TOA": Sine = Opposite/Hypotenuse, etc.).
Example Problem
- For a right triangle with sides 3, 4, and 5 (hypotenuse), let angle X be at one vertex.
- sin(X) = opposite/hypotenuse = 4/5.
- cos(X) = adjacent/hypotenuse = 3/5.
- tan(X) = opposite/adjacent = 4/3.
- Choosing a different reference angle (e.g., angle C) changes which sides are "opposite" and "adjacent":
- sin(C) = 3/5, cos(C) = 4/5, tan(C) = 3/4.
Key Terms & Definitions
- Right Triangle — a triangle with one 90° angle.
- Hypotenuse — the side opposite the right angle, longest side of a right triangle.
- Opposite side — the side across from the chosen angle.
- Adjacent side — the side next to the chosen angle (not the hypotenuse).
- Sine (sin) — ratio of the length of the opposite side to the hypotenuse.
- Cosine (cos) — ratio of the length of the adjacent side to the hypotenuse.
- Tangent (tan) — ratio of the length of the opposite side to the adjacent side.
Action Items / Next Steps
- Review trigonometry for special angles in the next lesson or video.
- Practice identifying opposite, adjacent, and hypotenuse for any chosen angle in right triangles.