note2 - Math021 week 2 Theorem 2.1 For every number x y...

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Math021, week 2 Theorem 2.1 For every number x , y sin( x + y ) = sin x cos y + cos x sin y sin( x - y ) = sin x cos y - cos x sin y cos( x + y ) = cos x cos y - sin x sin y cos( x - y ) = cos x cos y + sin x sin y tan( x + y ) = tan x +tan y 1 - tan x tan y tan( x - y ) = tan x - tan y 1+tan x tan y Corollary 2.2 For every number x , sin 2 x = 2 sin x cos x cos 2 x = cos 2 x - sin 2 x = 2 cos 2 x - 1 = 1 - 2 sin 2 x tan 2 x = 2 tan x 1 - tan 2 x Theorem 2.3 (cosine law) Let the lengths of the edges of a triangle be a , b , c . Its three angles are A , B , C such that the angle A is opposite to the edge with length a , etc. Then, c 2 = a 2 + b 2 - 2 ab cos C. Theorem 2.4 (Sine law) Let the lengths of the edges of a triangle be a , b , c . Its three angles are A , B , C such that the angle A is opposite to the edge with length a , etc. Then, sin A a = sin B b = sin C c . Definition 2.5 Let a > 0 . The function f ( x ) = a x for all x is called the exponential function with base a . Proposition 2.6 If a, b > 0 and x is a number, 1. a x a y = a x + y . 2. a x a y = a x - y .
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