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goniometric formulas∗
Wkbj79†
2013-03-21 22:42:21
The word goniometric (from Greek γωνı́α “angle” and µετ %ικóς “measuring”) concerns the trigonometric functions and their mutual connections. There
are a great amount of formulas involving these functions (usually for real arguments).
1. Pythagorean identities
• sin2 x + cos2 x = 1
• tan2 x + 1 = sec2 x
• 1 + cot2 x = csc2 x
2. Fractional identities
sin x
cos x
cos x
cot x =
sin x
1
cot x =
tan x
1
tan x =
cot x
1
csc x =
sin x
1
sec x =
cos x
• tan x =
•
•
•
•
•
3. Formulas involving radicals
• sin x = ± √
tan x
1 + tan2 x
1
• cos x = ± √
1 + tan2 x
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4. Weierstrass substitution formulas and related formula for tan x
x
2 tan
2 • sin x =
2 x
1 + tan
2
2 x
1 − tan
2
• cos x =
2 x
1 + tan
2
x
2 tan
2 • tan x =
2 x
1 − tan
2
5. Trigonometric functions of a purely imaginary number
• sin(ix) = i sinh x
• cos(ix) = cosh x
• tan(ix) = i tanh x
• cot(ix) = i coth x
• csc(ix) = i csch x
• sec(ix) = sech x
6. Addition formulas and subtraction formulas
• sin(x ± y) = sin x cos y ± cos x sin y
• cos(x ± y) = cos x cos y ∓ sin x sin y
tan x ± tan y
• tan(x ± y) =
1 ∓ tan x tan y
7. Formulas for trigonometric functions of a complex number
• sin(x + iy) = sin x cosh y + i cos x sinh y
• cos(x + iy) = cos x cosh y − i sin x sinh y
tan x + i tanh y
• tan(x + iy) =
1 − i tan x tanh y
8. Complement formulas
π
• sin
− x = cos x
2
π
• cos
− x = sin x
2
π
• tan
− x = cot x
2
9. Supplement formulas
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• sin(π − x) = sin x
• cos(π − x) = − cos x
• tan(π − x) = − tan x
10. Explement formulas
• sin(2π − x) = − sin x
• cos(2π − x) = cos x
• tan(2π − x) = − tan x
11. Opposite angle formulas
• sin(−x) = − sin x
• cos(−x) = cos x
• tan(−x) = − tan x
12. Periodicity formulas
• sin(x + 2π) = sin x
• cos(x + 2π) = cos x
• tan(x + π) = tan x
13. Double angle formulas
• sin(2x) = 2 sin x cos x
• cos(2x) = cos2 x − sin2 x = 2 cos2 x − 1 = 1 − 2 sin2 x
2 tan x
• tan(2x) =
1 − tan2 x
14. Triple angle formulas
• sin(3x) = 3 sin x − 4 sin3 x = (4 cos2 x − 1) sin x
• cos(3x) = 4 cos3 x − 3 cos x = (1 − 4 sin2 x) cos x
• tan(3x) =
3 tan x − tan3 x
1 − 3 tan2 x
15. Half angle formulas
r
x
1 − cos x
• sin
=±
2
2
r
x
1 + cos x
=±
• cos
2
2
r
x
sin x
1 − cos x
1 − cos x
• tan
=
=
=±
2
1 + cos x
sin x
1 + cos x
3
16. Prosthaphaeresis formulas
x+y
x−y
• sin x + sin y = 2 sin
cos
2
2
x−y
x+y
• sin x − sin y = 2 sin
cos
2
2
x+y
x−y
• cos x + cos y = 2 cos
cos
2
2
x+y
x−y
• cos x − cos y = −2 sin
sin
2
2
17. Product formulas
cos(x − y) − cos(x + y)
2
sin(x + y) − sin(x − y)
• cos x sin y =
2
cos(x − y) + cos(x + y)
• cos x cos y =
2
• sin x sin y =
18. Other sums and differences
sin(x ± y)
cos x cos y
sin(y ± x)
• cot x ± cot y =
sin x sin y
√
π
π
√
± x = 2 cos
∓x
• cos x ± sin x = 2 sin
4
4
• tan x ± tan y =
19. Linearization formulas
• Second power
1 − cos(2x)
2
1 + cos(2x)
2
– cos x =
2
1 − cos(2x)
2
– tan x =
1 + cos(2x)
– sin2 x =
• Third power
3 sin x − sin(3x)
4
3 cos x + cos(3x)
3
– cos x =
4
3 sin x − sin(3x)
3
– tan x =
3 cos x + cos(3x)
– sin3 x =
4
• Fourth power
cos(4x) − 4 cos(2x) + 3
8
cos(4x)
+
4
cos(2x) + 3
– cos4 x =
8
cos(4x) − 4 cos(2x) + 3
– tan4 x =
cos(4x) + 4 cos(2x) + 3
– sin4 x =
20. Recursion formulas
• sin[(n+1)x] = 2 cos x sin(nx) − sin[(n−1)x]
• cos[(n+1)x] = 2 cos x cos(nx) − cos[(n−1)x]
21. Exponential formulas
• eix = cos x + i sin x
• e−ix = cos x − i sin x
eix + e−ix
• cos x =
2
eix − e−ix
• sin x =
2i
eix − e−ix
• tan x =
i(eix + e−ix )
22. Some special formulas
•
•
•
•
r
π cos x + sin x
1 + sin 2x
tan x+
=
=±
4
cos x − sin x
1 − sin 2x
x π
+
tan x + sec x = tan
2
4
x±y
sin x ± sin y
cos y − cos x
tan
=
=
2
cos x + cos y
sin x ∓ sin y
x+y
x−y
cos y − cos x
tan
tan
=
2
2
cos y + cos x
• sin(x + y) sin(x − y) = sin2 x − sin2 y
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