Summary Complete Cheat Sheet to Trigonometry

Part III. Circular Functions and Unit Circle
I. Reciprocal Identities
sin(x)=1/csc(x)
cos(x)=1/sec(x)
tan(x)=sin(x)/cos(x)
cot(x)=1/tan(x)=cos(x)/sin(x)
csc(x)=1/sin(x)
sec(x)=1/cos(x)

II. Pythagorean Identities
sin2(x)+cos2(x)=1
1+tan2(x)=sec2(x)
1+cot2(x)=csc2(x)

III. Odd and Even Identities
y=sin(x) odd y=csc(x) odd
y=cos(x) even y=sec(x) even
y=tan(x) odd y=cot(x) odd

IV. Cofunction Identities
Case 1: π+x Case 2: π-x Case 3: 2π+x Case 4: 2π-x

sin(π+x)=-sin(x) sin(π-x)=sin(x) sin(2π+x)=sin(x) sin(2π-x)=-sin(x)
cos(π+x)=-cos(x) cos(π-x)=-cos(x) cos(2π+x)=cos(x) cos(2π-x)=cos(x)
tan(π+x)=tan(x) tan(π-x)=-tan(x) tan(2π+x)=tan(x) tan(2π-x)=-tan(x)
csc(π+x)=-csc(x) csc(π-x)=csc(x) csc(2π+x)=1/sin(x)=csc(x) csc(2π-x)=-1/sin(x)=-csc(x
sec(π+x)=-sec(x) sec(π-x)=-sec(x) sec(2π+x)=1/cos(x)=sec(x) sec(2π-x)=1/cos(x)=sec(x)
cot(π+x)=cot(x) cot(π-x)=-cot(x) cot(2π+x)=1/tan(x)=cot(x cot(2π-x)=-1/tan(x)=-cot(
) )

Case 5: π/2+x Case 6: π/2-x Case 7: 3π/2+x Case 8:

, sin(π/2+x)=cos(x) sin(π/2-x)=cos(x) sin(3π/2+x)=-cos(x) sin(3π/2-x)=-cos(x)
cos(π/2+x)=-sin(x) cos(π/2-x)=sin(x) cos(3π/2+x)=sin(x) cos(3π/2-x)=-sin(x)
tan(π/2+x)=-1/tan(x)=-cot(x tan(π/2-x)=cot(x) tan(3π/2+x)=-1/tan(x) tan(3π/2-x)=1/tan(x)
) csc(π/2-x)=sec(x) csc(3π/2+x)=-1/cos(x) csc(3π/2-x)=-1/cos(x)
csc(π/2+x)=1/cos(x)=sec(x) sec(π/2-x)=csc(x) sec(3π/2+x)=1/sin(x) sec(3π/2-x)=-1/sin(x)
sec(π/2+x)=-1/sin(x)=-csc(x) cot(π/2-x)=tan(x) cot(3π/2+x)=-tan(x) cot(3π/2-x)=tan(x)
cot(π/2+x)=-tan(x)


Exact Value Conversion
sin(x) cos(x) tan(x) csc(x) sec(x) cot(x)

0 0 1 0 undefined 1 undefined

π/6 1/2 √3/2 √3/3 2 2√3/3 √3

π/4 √2/2 √2/2 1 √2 √2 1

π/3 √3/2 1/2 √3 2√3/3 2 √3/3

π/2 1 0 undefined 1 undefined 0

π 0 -1 0 undefined -1 undefined

3π/2 -1 0 undefined -1 undefined 0

2π 0 1 0 undefined 1 undefined


Right Triangle Conversions
sin(x)=o/h
cos(x)=a/h
tan(x)=o/a
csc(x)=h/o
sec(x)=h/a
cot(x)=a/o