Summary of the Lecture
In today’s lecture, Professor Paulo Pereira introduced the basics of trigonometry in right-angled triangles. The session covered the properties and elements of right-angled triangles, named the sides of these triangles, discussed the internal angle sum, and introduced trigonometric ratios through the sine, cosine, and tangent functions.
Key Points from the Lecture
1. Understanding Right-Angled Triangles
- A right-angled triangle is defined by having one angle of 90 degrees, represented by a small square.
- The triangle receives its name because it is effectively half of a rectangle split along the diagonal.
- Sides of the right-angled triangle:
- Hypotenuse: The side opposite the 90 degree angle and the longest side.
- Legs (Catheti): The other two sides, called the catheti.
2. Other Angles in a Right-Angled Triangle
- The sum of internal angles in any triangle is 180 degrees.
- If one angle is 90 degrees, the other two must sum up to 90 degrees and thus are both acute.
3. The Pythagorean Theorem
- Equation: (hypotenuse^2 = leg_1^2 + leg_2^2)
- This fundamental theorem allows for the determination of any side of the triangle when the other two sides are known.
4. Trigonometric Ratios
- Sine, Cosine, and Tangent are key trigonometric ratios, defined as:
- Sine of an angle: Opposite leg / Hypotenuse
- Cosine of an angle: Adjacent leg / Hypotenuse
- Tangent of an angle: Opposite leg / Adjacent leg
Explanation using mnemonic "SohCahToa":
- Soh (Sine): Opposite/Hypotenuse
- Cah (Cosine): Adjacent/Hypotenuse
- Toa (Tangent): Opposite/Adjacent
5. Practical Application
- Demonstration through specific triangle calculations, showing how to use the Pythagorean theorem to find missing sides and then calculating the trigonometric ratios.
Example Calculation:
- Given: Hypotenuse = 21, one leg = 12
- Find other leg using Pythagorean theorem:
- (X^2 = 21^2 - 12^2 = 256 \Rightarrow X = 16)
- Calculate Sine, Cosine, and Tangent for an angle:
- Sine: Opposite leg (16) / Hypotenuse (21)
- Cosine: Adjacent leg (12) / Hypotenuse (21)
- Tangent: Opposite leg (16) / Adjacent leg (12)
6. Additional Exercise
- Example provided on calculating unknown sides using trigonometric sine ratio for angles such as 30 degrees, demonstrating the practical application in solving real-world problems using trigonometry.
Conclusion
The lecture emphasized the foundational knowledge necessary to understand more complex trigonometric functions and solve geometrical problems related to right-angled triangles.