Trigonometric Formulas

Trigonometric Formulas
Aaron Peterson
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Definitions
1. tan(x) =
sin(x)
cos(x)
2. sec(x) =
1
cos(x)
3. csc(x) =
1
sin(x)
4. cot(x) =
1
tan(x)
=
cos(x)
sin(x)
Pythagorean
1. sin2 (x) + cos2 (x) = 1
2. 1 + cot2 (x) = csc2 (x)
3. tan2 (x) + 1 = sec2 (x)
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Angle Sum and Difference
1. sin(φ ± θ) = sin(φ) cos(θ) ± cos(φ) sin(θ)
2. cos(φ ± θ) = cos(φ) cos(θ) ∓ sin(φ) sin(θ)
3. tan(φ ± θ) =
4
5
tan(φ)±tan(θ)
1∓tan(φ) tan(θ)
Combination With Different Arguments
1. sin(mx) sin(nx) =
1
2
(cos([m − n]x) − cos([m + n]x))
2. sin(mx) cos(nx) =
1
2
(sin([m − n]x) + sin([m + n]x))
3. cos(mx) cos(nx) =
1
2
(cos([m − n]x) + cos([m + n]x))
Power-Reducing
1. sin2 (θ) =
1−cos(2θ)
2
2. cos2 (θ) =
1+cos(2θ)
2
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Half-Angle
q
1. sin( θ2 ) = ± 1−cos(θ)
2
q
2. cos( θ2 ) = ± 1+cos(θ)
2
3. tan( θ2 ) = csc(θ) − cot(θ)
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Multiple Angle Formulae
1. sin(2θ) = 2 sin(θ) cos(θ)
2. cos(2θ) = cos2 (θ) − sin2 (θ)
2 tan(θ)
1−tan2 (θ)
3. tan(2θ) =
4. sin(nθ) =
Pn
n
k
5. cos(nθ) =
Pn
n
k
Where
n
k
k=0
=
k=0
cosk (θ) sinn−k (θ) sin
cosk (θ) sinn−k (θ) cos
1
2 (n
1
2 (n
− k)π
− k)π ,
n!
k! (n−k)! .
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