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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 ∗ hGoniometricFormulasi created: h2013-03-21i by: hWkbj79i version: h39293i Privacy setting: h1i hTopici h33B10i h26A09i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1 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 2 • 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 5