Trigonometry and Geometry

Equilateral Triangles

  • All three sides are the same length.
  • All three angles are the same measure.
  • Since the sum of angles in a triangle is 180^{\circ}, each angle in an equilateral triangle measures 60^{\circ}.

30-60-90 Triangles

  • Formed by dropping a perpendicular from one vertex to the midpoint of the opposite side in an equilateral triangle.
  • If the side length of the original equilateral triangle is 1, then:
    • The hypotenuse of the 30-60-90 triangle is 1.
    • The shorter leg (opposite the 30^{\circ} angle) is \frac{1}{2}.
    • The longer leg (opposite the 60^{\circ} angle) can be found using the Pythagorean theorem:
      h^2 + (\frac{1}{2})^2 = 1^2
      h^2 + \frac{1}{4} = 1
      h^2 = \frac{3}{4}
      h = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2}
  • Key measurements:
    • Hypotenuse: 1
    • Short leg (opposite 30^{\circ}): \frac{1}{2}
    • Long leg (opposite 60^{\circ}): \frac{\sqrt{3}}{2}

Isosceles Right Triangles (45-45-90 Triangles)

  • Isosceles means two sides are of equal length.
  • In an isosceles right triangle, the two legs are the same length, and one angle is 90^{\circ}.
  • The other two angles must be equal and add up to 90^{\circ}, making each 45^{\circ}.
  • If the hypotenuse is 1, we can find the length of the legs (denoted as a) using the Pythagorean theorem:
    a^2 + a^2 = 1^2
    2a^2 = 1
    a^2 = \frac{1}{2}
    a = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}}
  • Rationalizing the denominator:
    \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}
  • Key measurements:
    • Hypotenuse: 1
    • Legs (opposite 45^{\circ}): \frac{\sqrt{2}}{2}

Unit Circle

  • A circle with a radius of 1 unit, centered at the origin.
  • Equation: x^2 + y^2 = 1
  • Key points on the unit circle:
    • (1, 0) (0 degrees)
    • (0, 1) (90 degrees)
    • (-1, 0) (180 degrees)
    • (0, -1) (270 degrees)

Relationship Between Special Right Triangles and the Unit Circle

  • 30-60-90 triangle:
    • Placing a 30-60-90 triangle on the unit circle with the 30^{\circ} angle at the origin:
      • The coordinates of the point where the hypotenuse intersects the unit circle are (\frac{\sqrt{3}}{2}, \frac{1}{2}).
    • Placing a 30-60-90 triangle on the unit circle with the 60^{\circ} angle at the origin:
      • The coordinates of the point where the hypotenuse intersects the unit circle are (\frac{1}{2}, \frac{\sqrt{3}}{2}).
  • 45-45-90 triangle:
    • Placing a 45-45-90 triangle on the unit circle with the 45^{\circ} angle at the origin:
      • The coordinates of the point where the hypotenuse intersects the unit circle are (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}).

Reflections and Symmetry

  • Reflecting points across the y-axis changes the sign of the x-coordinate but not the y-coordinate.
  • Reflecting points across the x-axis changes the sign of the y-coordinate but not the x-coordinate.
  • Reflecting across both axes changes the sign of both coordinates.

Key Angles and Coordinates on the Unit Circle

  • 30^{\circ}: (\frac{\sqrt{3}}{2}, \frac{1}{2})
  • 45^{\circ}: (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})
  • 60^{\circ}: (\frac{1}{2}, \frac{\sqrt{3}}{2})
  • By using reflections, we can determine the coordinates of points on the unit circle for angles in all four quadrants.

Important Considerations

  • The x and y coordinates of any point on the unit circle must satisfy the equation x^2 + y^2 = 1.
  • Being able to quickly recall these key angles and coordinates is essential for understanding trigonometry.