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December 2022

# trigonometric circle and trigonometric identities

Math cheat-sheet: The angles 0, 30', 45', 60' and 90' have all special properties:

trigonometric circle, values on the outside are: {cos,sin}

-0'30'=π/6
=0.5236
45'=π/4
=0.7854
60'=π/3
=1.0472
90'=π/2
=1.5708
sinsqrt(0)/2
=0
sqrt(1)/2
=0.5
sqrt(2)/2
=0.7071
sqrt(3)/2
=0.8660
sqrt(4)/2
=1
cossqrt(4)/2
=1
sqrt(3)/2
=0.8660
sqrt(2)/2
=0.7071
sqrt(1)/2
=0.5
sqrt(0)/2
=0
tan=sin/cos0sqrt(3)/3
=0.5774
1sqrt(3)
=1.7321
undefinded

## Trigonometric identities

1. sin(x)=+/- sqrt(1-cos2x)
2. cos(x)=+/- sqrt(1-sin2x)
3. sin(-x)=-sin(x)
4. cos(-x)=cos(x)
5. sin(x +/- π/2)=+/- cos(x)
6. cos(x +/- π/2)=-/+ sin(x)
7. sin(a +/- b) = sin(a)*sin(b) +/- cos(a)*cos(b)
8. cos(a +/- b) = cos(a)*cos(b) -/+ sin(a)*sin(b)
9. sin(2x)=2sin(x)cos(x)=(sin(x)+cos(x))2
10. cos(2x)=cos2(x)-sin2(x)=2cos2(x) -1
11. cos(a)cos(b)=[cos(a-b)+cos(a+b)]/2
12. sin(a)sin(b)=[cos(a-b)-cos(a+b)]/2
13. sin(a)cos(b)=[sin(a+b)-sin(a-b)]/2

## Trigonometric functions and complex exponential functions

e with an imaginary exponent is a periodic function:
1. e=cos(θ) + i sin(θ)
2. sin(θ)=( e - e-iθ )/2i
3. cos(θ)=( e + e-iθ )/2