Calculus

From Trigonometry​(TRIGO-1)

• Trigonometry: "Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. It's a fundamental concept that plays a crucial role in various fields, including engineering, physics, astronomy, and navigation. Trigonometry allows us to solve problems involving distances, heights, and angles."

• Angle: "Angles are fundamental to trigonometry. An angle is formed by the rotation of a line segment around a fixed point called the vertex."

• Degrees: "Degrees are a common unit for measuring angles. A full circle is divided into 360 degrees."

• Radians: "Radians are another unit for measuring angles. One radian is the angle subtended at the center of a circle by an arc equal in length to the radius of the circle. A full circle measures 2π radians."

• Standard Position: "In standard position, the initial side of an angle coincides with the positive x-axis, and the vertex is at the origin."

• Quadrants: "The unit circle is divided into four quadrants, numbered I through IV, starting from the upper right quadrant and moving counterclockwise."

• Unit Circle: "A circle with a radius of 1 unit, centered at the origin of a coordinate plane. It is commonly used in trigonometry to define the trigonometric functions for all real numbers."

• Pythagorean Theorem: "The Pythagorean theorem is a fundamental relationship in right-angled triangles. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides."

• Trigonometric Functions (Sine, Cosine, Tangent): "Trigonometric functions relate the angles of a right-angled triangle to the lengths of its sides. The three primary trigonometric functions are sine, cosine, and tangent. These functions are defined as ratios of the sides of a right-angled triangle relative to a specific angle."

• Special Angles: "There are certain special angles in trigonometry that have specific and easily remembered trigonometric values. These angles are 0°, 30°, 45°, 60°, and 90°."

• Sine Law: "The sine law states that the ratio of a side to the sine of its opposite angle is constant for all sides and angles of a triangle."

• Cosine Law: "The cosine law relates the square of one side of a triangle to the squares of the other two sides and the cosine of the included angle."

• Angle of Elevation: "The angle of elevation is the angle formed between the horizontal line and the line of sight when an observer is looking upward at an object above the horizontal."

• Angle of Depression: "The angle of depression is the angle formed between the horizontal line and the line of sight when an observer is looking downward at an object below the horizontal."

From Trigonometry​(TRIGO-2)

• Basic Trigonometric Ratios: Lists the standard trigonometric ratios (sine, cosine, tangent).

• Reciprocal Identities: Lists identities involving reciprocal functions such as secant and cosecant.

• Pythagorean Identities: Lists identities derived from the Pythagorean theorem (e.g., sin²(x) + cos²(x) = 1).

• Angle Sum and Difference Identities: "Formulas for trigonometric functions of angle sums/differences."

• Double Angle Identities: "Identities for trigonometric functions of twice an angle."

• Half Angle Identities: "Identities for trigonometric functions of half an angle."

• Product-to-Sum and Sum-to-Product Formulas: "Convert products of trigonometric functions into sums or vice versa."

• Trigonometric Equations: "Trigonometric equations are mathematical statements that equate trigonometric functions to a value or another function."

• Inverse Trigonometric Equations: "These involve arcsine, arccosine, and arctangent functions."

• General Solutions in Trigonometry: "Trigonometric functions are periodic, which means solutions will often involve a general formula that includes k, an integer representing the number of complete cycles."

• Proving Trigonometric Identities: "Proving trigonometric identities involves demonstrating that one side of an equation can be transformed into the other side using known identities, algebraic manipulations, and sometimes geometric interpretations."J

From Plane Geometry​(Plane Geometry)

• Polygon: "A polygon is a closed plane figure formed by line segments."

  • Side or Edge: "The side or edge of a polygon is one of the line segments that make up the polygon."

  • Vertex: "The vertices of a polygon are the end points of each side of the polygon."

  • Diagonal: "A diagonal of a polygon is a line segment joining two non-adjacent vertices of the polygon."

  • Interior Angle: "The angle formed by two adjacent sides inside the polygon."

  • Exterior Angle: "An angle that is adjacent to and supplementary to an interior angle of the polygon."

• Types of Polygons:

  • Equiangular Polygon: "A polygon is equiangular if all of its angles are congruent."

  • Equilateral Polygon: "A polygon is equilateral if all of its sides are equal."

  • Regular Polygon: "Regular polygons are both equiangular and equilateral."

  • Irregular Polygon: "A polygon that is neither equiangular nor equilateral is said to be an irregular polygon."

  • Convex Polygon: "Every interior angle is less than 180°. If a line is drawn through the convex polygon, the line will intersect at most two sides."

  • Concave Polygon: "A concave polygon has at least one interior angle that measures more than 180°. If a line is drawn through a concave polygon the line may intersect more than two sides."

• Similar Polygons: "Two polygons are similar if their corresponding interior angles are congruent and their corresponding sides are proportional."

• Perimeter of a Polygon: "To find a perimeter of a polygon, add the lengths of the sides of the polygon."

• Central Angle: "The angle that is opposite a side of a regular polygon is a central angle θ of the polygon."

• Apothem: "The altitude of the isosceles triangles that can be formed from a regular polygon is the apothem of the regular polygon."

From Solid Geometry​(SOLID GEOMETRY)

• Solid Geometry: "is the study of three-dimensional figures, also called solids or 3D shapes. These shapes have depth, in addition to length and width, unlike plane geometry, which only deals with two-dimensional shapes."

• Volume: "The amount of space a solid object occupies, typically measured in cubic units (e.g., cubic meters, cubic centimeters)."

• Surface Area: "The total area of all the surfaces or faces of a three-dimensional object."

• Faces, Edges, and Vertices:

  • Faces: "The flat surfaces of a 3D shape."

  • Edges: "The line segments where two faces meet."

  • Vertices: "The points where edges meet."

• Cube: "A six-faced polyhedron where all faces are squares and all edges are of equal length."

• Rectangular Prism: "A polyhedron with six rectangular faces, also known as a cuboid."

• Cylinder: "A solid with two parallel circular bases connected by a curved surface."

• Sphere: "A perfectly round solid where every point on the surface is equidistant from the center."

• Cone: "A solid with a circular base and a single vertex, connected by a curved surface."