Math 1122 Lecture 5 Study Notes

Math 1122 Lecture 5 Study Notes

Recap: Geometry

Key Concepts Related to Triangles

• Triangles: A polygon with three edges and three vertices.

• Types of Triangles:

  • Right Triangles: Have one angle measuring 90 degrees.

  • Properties of Right Triangles:

    • The hypotenuse is the longest side, often denoted as C.

    • The relationship between the sides is defined by the Pythagorean theorem: $a^2 + b^2 = c^2$.

  • Similar Triangles: Triangles that have the same angles but may differ in size.

  • Congruent Triangles: Triangles that have the same shape and size (same sides).

• Triangle Pointers:

  • For triangle sides: $a + b > c$, $a + c > b$, $b + c > a$ (triangle inequality theorem).

Properties of Area and Perimeter

• Perimeter: The sum of all sides of the triangle.

• Area: For a triangle, the formula is given as $Area = \frac{1}{2}bh$ where b is the base and h is the height of the triangle.

  • Notably, the height h must be defined clearly in relation to the base.

Special Triangle Measurements

• For any right triangle:

  • The sum of the angles is always $A + B + C = 180°$.

  • If one angle is $90°$, then the sum of the other two angles is $90°$ (i.e., $B + C = 90°$ for a right triangle where $A$ is $90°$).

  • Corresponding Parameters:

  • Hypotenuse: C

Scaling and Graphing Right Triangles

• Right triangles can be scaled down relative to the hypotenuse of 1, allowing the creation of a graph where:

  • Let a be represented as x and b as y.

  • Then the circular equation is $x^2 + y^2 = 1$.

  • This equation forms a unit circle, where each right triangle is determined by one angle.

  • Starting angle chosen at (1,0) and rotated counterclockwise.

Angles' Relation to Triangles

• Exploring angles beyond basics:

  • Triangle at various degree measures:

  • Examples include measurements such as (40°), (45°), and (765°).

  • Understanding periodicity:

  • For example, angle 765° can be simplified through $360° + 360° + 45°$, giving us the same triangle (because angles are periodic).

Periodic Functions

• Generalization on periodic functions in trigonometry:

  • The concept that certain functions repeat their outputs over defined intervals. Period refers to the distance between repetitions in a periodic function.

• Two important periodic functions in trigonometry:

  • Sine Function: $Sin(x)$

  • Cosine Function: $Cos(x)$

• Characteristics of Sin(x) and Cos(x):

  • Both functions have a period of $2\pi$.

  • The midline is at $y=0$.

  • The amplitude (height of wave) is 1.

• Reason for $2\pi$: The value is related to circles, specifically the process of “gluing” construction on real numbers, reflecting their circular nature and continuity.

Conclusion

• The discussion on triangles leads into periodic functions, particularly through the relationships defined in geometry and the innate properties of angles in regards