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Math Formulas That Will Be Useful for Life Math Formulas That Will Be Useful for Life Coordinate geometry: A pair of axes that intersect perpendicularly in a plane allows any point on that plane to be represented by a set of real numbers. The intersection of the axes is (0, 0), which is called the origin. The positions in the horizontal and vertical directions are represented by x and y, respectively. A straight line can be represented by the equation y = mx + c, where m is the slope of the line. This line intersects the y-axis at (0, c) and the x-axis at (–c/m, 0). The equation of a vertical line is x = k, where x is a constant value. The line that passes through the point (x0, y0) with a slope of n is given by the equation y – y0 = n(x – x0). If a line is perpendicular to a line with slope n, then its slope is –1/n. The line passing through the points (x1, y1) and (x2, y2) is given by y = (y2 – y1)/(x2 – x1)(x – x2) + y2, provided that x1 ≠ x2. If the slopes of two lines are m and n respectively, then the angle θ between them satisfies tanθ = (m – n)/(1 + mn). A circle with radius r and center at (a, b) is represented by the equation (x – a)² + (y – b)² = r². Coordinates in three-dimensional space are similar to those in two-dimensional space, with the addition of a z-axis; for example, a sphere with radius r and center at (a, b, c) is represented by the equation (x–a)² + (y–b)² + (z–c)² = r². The general form of a plane in three-dimensional space is ax + by + cz = d. Trigonometry: A right triangle with side lengths a, b, and c, where one of the angles is θ. Its six trigonometric functions are: sine, cosine, tangent, cosecant, secant, and cotangent. sinθ = b/c cosθ = a/c tanθ = b/a cscθ = c/b secθ = c/a cotθ = a/b. If the radius of the circle is 1, then its sine and cosine values correspond to the height and base of a right triangle, respectively. a = cosθ b = sinθ. According to the Pythagorean theorem, we know that a2 + b2 = c2. Therefore, for any angle θ on a circle, we can derive the following identity: cos2θ + sin2θ = 1. Trigonometric identities. Based on the definitions given on previous pages, the following identities hold: tanθ = sinθ/cosθ, cotθ = cosθ/sinθ, secθ = 1/cosθ, cscθ = 1/sinθ. By dividing both sides of cos2θ + sin2θ = 1 by cos2θ and sin2θ respectively, we get: sec2θ – tan2θ = 1, and csc2θ – cot2θ = 1. For negative angles, the six trigonometric functions are as follows: sin(–θ) = –sinθ, csc(–θ) = –cscθ; cos(–θ) = cosθ, sec(–θ) = secθ; tan(–θ) = –tanθ, cot(–θ) = –cotθ. When two angles are added, the sum formula is used: sin(α + β) = sinαcosβ + cosαsinβ, cos(α + β) = cosαcosβ – sinαsinβ, tan(α + β) = tanα + tanβ / (1 – tanαtanβ). For double or triple angles, the angle-doubling formulas are applied: sin2α = 2sinαcosα, sin3α = 3sinαcos2α – sin3α; cos2α = cos2α – sin2α, cos3α = 4cos3α – 3cosα; tan2α = 2tanα / (1 – tan2α), tan3α = 3tanα – tan3α / (1 – 3tan2α). Two-dimensional shapes. Below are some formulas for the perimeter and area of two-dimensional shapes. Circle: Radius = r Diameter d = 2r. Circumference = 2πr = πd. Area = πr² (π = 3.1415926……) Ellipse: Area = πab, where a and b represent half of the minor axis and major axis respectively. Rectangle: Area = ab; Perimeter = 2a + 2b. Parallelogram: Area = bh = ab sinα; Perimeter = 2a + 2b. Trapezoid: Area = 1/2h (a + b); Perimeter = a + b + h (secα + secβ). Regular n-gon: Area = 1/2nb² cot(180°/n); Perimeter = nb. Quadrilateral (i): Area = 1/2ab sinα; Quadrilateral (ii): Area = 1/2 (h1 + h2)b + ah1 + ch2. Three-dimensional shapes: Below are the formulas for the volume and surface area (including the base) of three-dimensional solids. Sphere: Volume = 4/3πr³, Surface Area = 4πr². Cube: Volume = abc, Surface Area = 2(ab + ac + bc). Cylinder: Volume = πr²h, Surface Area = 2πrh + 2πr². Cone: Volume = 1/3πr²h, Surface Area = πr√(r² + h²) + πr². Triangular pyramid: If the base area is A, then Volume = 1/3Ah. Frustum: Volume = 1/3πh(a² + ab + b²), Surface Area = π(a + b)c + πa² + πb². Ellipsoid: Volume = 4/3πabc. Torus: Volume = 1/4π²(a + b)(b – a)², Surface Area = π²(b² – a²)