Morris Dovey’s Trigonometry Refresher

 
For any right triangle ABC:
[right triangle image]

where a and b denote the lengths of the legs oposite the acute angles A and B; and c denotes the hypoteneuse (the side opposite right angle C),

sin(A) = a/csin(B) = b/c
cos(A) = b/ccos(B) = a/c
tan(A) = a/b
       = sin(A)/cos(A)
tan(B) = b/a
       = sin(B)/cos(B)
sec(A) = c/b
       = 1/cos(A)
sec(B) = c/a
       = 1/cos(B)
csc(A) = c/a
       = 1/sin(A)
csc(B) = c/b
       = 1/sin(B)
cot(A) = b/a
       = 1/tan(A)
cot(B) = a/b
       = 1/tan(B)
 
sin(2·x) = 2·sin(x)·cos(x)
cos(2·x) = cos²(x) - sin²(x)
         = 2·cos²(x) - 1
         = 1 - 2·sin²(x)
tan(2·x) = 2·tan(x)/(1 - tan²(x))
 
sin(x/2) = ± sqrt((1 - cos(x))/2)
cos(x/2) = ± sqrt((1 + cos(x))/2)
tan(x/2) = ± sqrt((1 - cos(x))/(1 + cos(x)))  
         = sin(x)/(1 + cos(x))
         = (1 - cos(x))/sin(x)
         = (1 ± sqrt(1 + tan²(x))/tan(x)
         = tan(A)·sin(x)/(tan(x) + sin(x))
 
a/sin(A) = b/sin(B) = c/sin(C) (Law of Sines)
 
c² = a² + b² - 2·a·b·cos(C) (Law of Cosines)
b² = a² + c² - 2·a·c·cos(B)
a² = b² + c² - 2·b·c·cos(A)
 
(a - b)/(a + b) = tan((A-B)/2)/tan((A+B)/2) (Law of Tangents)
 
cos(x + y) = cos(x)·cos(y) - sin(x)·sin(y)
cos(x - y) = cos(x)·cos(y) + sin(x)·sin(y)
sin(x + y) = sin(x)·cos(y) + cos(x)·sin(y)
sin(x - y) = sin(x)·cos(y) - cos(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))
 
sin(-x) = -sin(x)
csc(-x) = -csc(x)
cos(-x) = cos(x)
sec(-x) = sec(x)
tan(-x) = -tan(x)
cot(-x) = -cot(x)

Copyright © 2003 Morris R Dovey

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