Geometry: Triangles and Angles

Triangles

• A triangle is a polygon with three sides and three angles.
• Types of triangles:
  • Right-angled triangle: one angle is 90 degrees (π/2 radians)
  • Oblique triangle: no right angles
  • Acute triangle: all angles are acute (less than 90 degrees)
  • Obtuse triangle: one angle is obtuse (greater than 90 degrees)
• Properties:
  • Angle sum: interior angles add up to 180 degrees (π radians)
  • Side lengths: can be calculated using trigonometric ratios

Angles

• Measured in degrees (°) or radians (rad)
• Types of angles:
  • Acute angle: less than 90 degrees (π/2 radians)
  • Right angle: exactly 90 degrees (π/2 radians)
  • Obtuse angle: greater than 90 degrees (π/2 radians) but less than 180 degrees (π radians)
  • Straight angle: exactly 180 degrees (π radians)
• Angle relationships:
  • Complementary angles: add up to 90 degrees (π/2 radians)
  • Supplementary angles: add up to 180 degrees (π radians)

Sine (sin)

• Defined as the ratio of the opposite side to the hypotenuse in a right-angled triangle
• sin(A) = opposite side / hypotenuse
• Range: -1 ≤ sin(A) ≤ 1
• Graph: sine wave with period 2π

Cosine (cos)

• Defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle
• cos(A) = adjacent side / hypotenuse
• Range: -1 ≤ cos(A) ≤ 1
• Graph: cosine wave with period 2π

Tangent (tan)

• Defined as the ratio of the opposite side to the adjacent side in a right-angled triangle
• tan(A) = opposite side / adjacent side
• Range: all real numbers
• Graph: tangent wave with period π

Note: These notes provide a concise overview of the key concepts and formulas in trigonometry, focusing on triangles, angles, and the three main trigonometric ratios: sine, cosine, and tangent.

Triangles

• A triangle has three sides and three angles.

Types of Triangles

• A right-angled triangle has one angle of 90 degrees (π/2 radians).
• An oblique triangle has no right angles.
• An acute triangle has all angles less than 90 degrees.
• An obtuse triangle has one angle greater than 90 degrees.

Properties of Triangles

• The interior angles of a triangle add up to 180 degrees (π radians).
• Side lengths of a triangle can be calculated using trigonometric ratios.

Angles

• Angles are measured in degrees (°) or radians (rad).

Types of Angles

• An acute angle is less than 90 degrees (π/2 radians).
• A right angle is exactly 90 degrees (π/2 radians).
• An obtuse angle is greater than 90 degrees (π/2 radians) but less than 180 degrees (π radians).
• A straight angle is exactly 180 degrees (π radians).

Angle Relationships

• Complementary angles add up to 90 degrees (π/2 radians).
• Supplementary angles add up to 180 degrees (π radians).

Sine (sin)

• Sine is the ratio of the opposite side to the hypotenuse in a right-angled triangle.
• The formula for sine is sin(A) = opposite side / hypotenuse.
• The range of sine is -1 ≤ sin(A) ≤ 1.
• The graph of sine is a sine wave with a period of 2π.

Cosine (cos)

• Cosine is the ratio of the adjacent side to the hypotenuse in a right-angled triangle.
• The formula for cosine is cos(A) = adjacent side / hypotenuse.
• The range of cosine is -1 ≤ cos(A) ≤ 1.
• The graph of cosine is a cosine wave with a period of 2π.

Tangent (tan)

• Tangent is the ratio of the opposite side to the adjacent side in a right-angled triangle.
• The formula for tangent is tan(A) = opposite side / adjacent side.
• The range of tangent is all real numbers.
• The graph of tangent is a tangent wave with a period of π.

Learn about the types and properties of triangles, including right-angled, oblique, acute, and obtuse triangles, as well as angle measurements and trigonometric ratios.