Geometric Bites

These workings use compound angle identities to derive double angle, triple angle, and related trigonometric identities. Compound Angles, Extras 1. Deriving sin(2θ) sin(θ + θ) = sinθ cosθ + cosθ sinθ = 2sinθ cosθ = sin(2θ) 2. Deriving cos(2θ) cos(θ + θ) = cosθ cosθ - sinθ sinθ = cos 2 θ - sin 2 θ = cos(2θ) 3. Deriving cos(2θ) = 2cos²θ - 1 cos(2θ) = cos 2 θ - sin 2 θ = cos 2 θ - (1 - cos 2 θ) = cos 2 θ - 1 + cos 2 θ = 2cos 2 θ - 1 4. Deriving cos(2θ) = 1 - 2sin²θ cos(2θ) = cos 2 θ - sin 2 θ = 1 - sin 2 θ - sin 2 θ = 1 - 2sin 2 θ 5. Deriving sin(3θ) sin(2θ + θ) = sin(2θ)cosθ + cos(2θ)sinθ = 2sinθ cosθ cosθ + (1 - 2sin 2 θ)sinθ = 2sinθ cos 2 θ + sinθ - 2sin 3 θ = sinθ(2cos 2 θ + 1) - 2sin 3 θ = sinθ(2(1 - sin 2 θ) + 1) - 2sin 3 θ = sinθ(2 - 2sin 2 θ + 1) - 2sin 3 θ = sinθ(3 - 2sin 2 θ) - 2sin 3 θ = 3sinθ - 2sin 3 θ - 2sin 3 θ = 3sinθ - 4sin 3 θ 6. Deriving cos(3θ) cos(2θ + θ) = cos2θ cosθ - sin2...

These workings derive the laws of base-10 logarithms from exponent laws by converting between exponential form and logarithmic form. Assume a > 0 , b > 0 , and n ≠ 0 . Product Rule log(ab) = log a + log b Let 10 x = a Let 10 y = b Therefore: log 10 a = x log 10 b = y 10 x 10 y = ab 10 x+y = ab Therefore: log 10 (ab) = x + y Substituting: log 10 (ab) = log 10 a + log 10 b Therefore: log(ab) = log a + log b Quotient Rule log(a / b) = log a - log b Let 10 x = a Let 10 y = b Therefore: log 10 a = x log 10 b = y 10 x / 10 y = a / b 10 x-y = a / b Therefore: log 10 (a / b) = x - y Substituting: log 10 (a / b) = log 10 a - log 10 b Therefore: log(a / b) = log a - log b Power Rule log(a n ) = n log a Let log(a n ) = x Therefore: 10 x = a n Taking the n-th root of both sides: (10 x ) 1/n = (a n ) 1/n 10 x/n = a Therefore: log 10 a = x / n n log 10 a = x Theref...

Parabolas in sight: The Clifton Suspension Bridge, Bristol, United Kingdom. A quadratic function is a function whose graph is a parabola (a U-shaped curve). One of the most useful ways to write a quadratic is in vertex form , because it shows the parabola’s turning point immediately. 1) The vertex form A quadratic function in vertex form is written as: f(x) = a(x - h) 2 + k This form is especially helpful because the values h and k tell you the vertex directly. 2) The vertex (turning point) The vertex is the point where the parabola changes direction. In vertex form: Vertex = (h, k) If the parabola opens up , the vertex is the lowest point (a minimum). If the parabola opens down , the vertex is the highest point (a maximum). 3) What the number a does The number a controls two key things: the direction the parabola opens, and how wide or narrow it is. a > 0 means the parabola opens up (U-shape). a < 0 means the parabola opens dow...