Geometry: Understanding Angles

Angles

• Measure of an angle: The amount of rotation from the initial side to the terminal side.
• Types of angles:
  • Acute angle: Less than 90 degrees.
  • Right angle: Exactly 90 degrees.
  • Obtuse angle: Greater than 90 degrees but less than 180 degrees.
  • Straight angle: Exactly 180 degrees.
  • Reflex angle: Greater than 180 degrees but less than 360 degrees.
• Angle notation: Typically denoted by Greek letters (e.g., θ, α, β) or by using the "∠" symbol.

Triangles

• Types of triangles:
  • Right triangle: One right angle (90 degrees).
  • Oblique triangle: No right angles.
  • Isosceles triangle: Two sides of equal length.
  • Equilateral triangle: All sides of equal length.
  • Scalene triangle: All sides of different lengths.
• Triangle properties:
  • Sum of angles: 180 degrees.
  • Pythagorean theorem: a^2 + b^2 = c^2 (for right triangles).

Sine (sin)

• Definition: The ratio of the opposite side to the hypotenuse.
• Formula: sin(θ) = opposite side / hypotenuse.
• Range: -1 ≤ sin(θ) ≤ 1.
• Graph: Periodic, with a period of 360 degrees.

Cosine (cos)

• Definition: The ratio of the adjacent side to the hypotenuse.
• Formula: cos(θ) = adjacent side / hypotenuse.
• Range: -1 ≤ cos(θ) ≤ 1.
• Graph: Periodic, with a period of 360 degrees.

Tangent (tan)

• Definition: The ratio of the opposite side to the adjacent side.
• Formula: tan(θ) = opposite side / adjacent side.
• Range: All real numbers.
• Graph: Periodic, with a period of 180 degrees.

Note: These notes provide a brief overview of the key concepts and formulas related to angles, triangles, and the three main trigonometric functions (sine, cosine, and tangent).

Angles

• Measure of an angle is the amount of rotation from the initial side to the terminal side.
• Angles can be acute (less than 90 degrees), right (exactly 90 degrees), obtuse (greater than 90 degrees but less than 180 degrees), straight (exactly 180 degrees), or reflex (greater than 180 degrees but less than 360 degrees).
• Angles are typically denoted by Greek letters (e.g., θ, α, β) or by using the "∠" symbol.

Triangles

• Triangles can be right (one right angle), oblique (no right angles), isosceles (two sides of equal length), equilateral (all sides of equal length), or scalene (all sides of different lengths).
• The sum of angles in a triangle is always 180 degrees.
• The Pythagorean theorem (a^2 + b^2 = c^2) applies to right triangles.

Sine (sin)

• Sine is the ratio of the opposite side to the hypotenuse in a right triangle.
• The formula for sine is sin(θ) = opposite side / hypotenuse.
• The range of sine values is -1 ≤ sin(θ) ≤ 1.
• The graph of sine is periodic with a period of 360 degrees.

Cosine (cos)

• Cosine is the ratio of the adjacent side to the hypotenuse in a right triangle.
• The formula for cosine is cos(θ) = adjacent side / hypotenuse.
• The range of cosine values is -1 ≤ cos(θ) ≤ 1.
• The graph of cosine is periodic with a period of 360 degrees.

Tangent (tan)

• Tangent is the ratio of the opposite side to the adjacent side in a right triangle.
• The formula for tangent is tan(θ) = opposite side / adjacent side.
• The range of tangent values is all real numbers.
• The graph of tangent is periodic with a period of 180 degrees.

Regular Polygons

Interior Angles

• The sum of interior angles in a regular polygon with n sides is (n-2) × 180°, which means that as the number of sides increases, the total sum of interior angles also increases.
• Each interior angle measures (n-2) × 180° / n, indicating that the measure of each interior angle decreases as the number of sides increases.
• As the number of sides (n) increases, the interior angle approaches 180°, meaning that the polygon approaches a straight line.

Exterior Angles

• The sum of exterior angles in a regular polygon is always 360°, regardless of the number of sides.
• Each exterior angle measures 360° / n, indicating that the measure of each exterior angle decreases as the number of sides increases.
• The exterior angle is supplementary to the interior angle, meaning they add up to 180°.

Apothem

• The apothem is the distance from the center of the polygon to one of its vertices, providing a key measure for calculating area and perimeter.
• The apothem is also the radius of the circle inscribed in the polygon, highlighting its importance in geometric calculations.
• The apothem is used to find the area and perimeter of the polygon, making it a crucial concept in polygon geometry.

Perimeter

• The perimeter of a regular polygon is the distance around the polygon, providing a measure of its overall size.
• The formula for the perimeter is P = n × side length, making it easy to calculate given the number of sides and side length.
• The perimeter can also be calculated using the formula P = 2n × apothem × tan(π/n), offering an alternative method for calculation.

Area

• The area of a regular polygon is given by the formula A = (n × side length × apothem) / 2, providing a straightforward method for calculation.
• The area can also be found using the formula A = (n × r^2 × sin(2π/n)) / 2, where r is the radius of the circumscribed circle, offering an alternative method.
• The area of a regular polygon is proportional to the number of sides and the square of the side length, highlighting the importance of these variables in area calculations.