M.Eng. René Schwarz Master of Engineering in Computer Science and Communication Systems (M.Eng.) Bachelor of Engineering in Mechatronics, Industrial and Physics Technology (B.Eng.) Memorandum № 3 Trigonometric Functions Note: All angles in this paper are given in radians, unless otherwise stated. 1 Trigonometric functions and their definitions . Other (Image Credit: [3]) Function Abbreviation Geometric Description Identities ( ) opposite a π https://creativecommons.org/licenses/by-nc/4.0/ − Sine sin(α) sin α = hypothenuse = c sin α = cos α 2 ), unless otherwise stated. ( ) (α) α = adjacent = b α = α + π Cosine cos cos hypothenuse c cos sin 2 ( ) (α) α = opposite = a α = sin α = π − α = 1 Tangent tan tan adjacent b tan cos α cot ( 2 ) cot α (α) α = adjacent = b α = cos α = π − α = 1 Cotangent cot cot opposite a cot sin α( tan) 2 tan α (α) α = hypothenuse = c α = π − α = 1 Secant sec sec adjacent b sec csc ( 2 ) cos α rene-schwarz.com ( hypothenuse c π − 1 Cosecant csc(α) csc α = opposite = a csc α = sec 2 α = sin α René Schwarz 2 Shorthands Name Shorthand Name Shorthand This document is subjectLicense to (CC the conditions BY-NC 4.0), whichof the can Creative be Commons Attribution-NonCommercial examined at 4.0 International ©2017 M.Eng. licenses may apply for particular contents or supplemental documents/files. − half coversed sine (hacoversine) hacoversin α = 1 sin α versed sine (versine) versin α = 2 sin2 α = 1 − cos α 2 2 half coversed cosine (hacovercosine) hacovercosin α = 1+sin α versed cosine (vercosine) vercosin α = 2 cos2 α = 1 + cos α 2 2 exterior secant (exsecant) exsec α = sec α − 1 coversed sine (coversine) coversin α = 1 − sin α exterior cosecant (excosecant) excsc α = csc α − 1 coversed cosine (covercosine) covercosin α = 1 + sin α α 1−cos α chord crd α = 2 sin 2 half versed sine (haversine) haversin α = 2 sin(πα) cbn 1−sin α sinus cardinalis (cardinal sine) sinc α = πα (normalized) half versed cosine (havercosine) havercosin α = 2 sin α sinc α = α (unnormalized) 3 Important values of the basic trigonometric functions 1 1 1 1 2 3 5 7 5 4 3 5 7 11 0 6 π 4 π 3 π 2 π 3 π 4 π 6 π π 6 π 4 π 3 π 2 π 3 π 4 π 6 π 2π 0◦ 30◦ 45◦ 60◦ 90◦ 120◦ 135◦ 150◦ 180◦ 210◦ 225◦ 240◦ 270◦ 300◦ 315◦ 330◦ 360◦ p p p p p p p p sin α 0 1 2 3 1 3 2 1 0 − 1 − 2 − 3 −1 − 3 − 2 − 1 0 p2 p2 2 2 2p 2p p2 p2 2 2 p2 p2 cos α 1 3 2 1 0 − 1 − 2 − 3 −1 − 3 − 2 − 1 0 1 2 3 1 p2 2 p2 p2 2 p2 p2 2 p2 p2 2 2p tan α 0 3 1 3 ∞ − 3 −1 − 3 0 3 1 3 ∞ − 3 −1 − 3 0 p3 p p p3 p3 p p p3 ∞ 3 − 3 − − ∞ 3 − 3 − − ∞ cot α 3 1 3 0 3 1 3 3 1 3 0 3 1 3 Mnemonics for sine and cosine: 1 1 1 1 2 3 5 7 5 4 3 5 7 11 https://goo.gl/XB85LZ 0 6 π 4 π 3 π 2 π 3 π 4 π 6 π π 6 π 4 π 3 π 2 π 3 π 4 π 6 π 2π Errors, comments or ideas regarding! this paper? 0◦ 30◦ 45◦ 60◦ 90◦ 120◦ 135◦ 150◦ 180◦ 210◦ 225◦ 240◦ 270◦ 300◦ 315◦ 330◦ 360◦ p p p p p p p p p p p p p p p p p sin α 0 1 2 3 4 3 2 1 0 − 1 − 2 − 3 − 4 − 3 − 2 − 1 0 p2 p2 p2 p2 p2 2p 2p 2p 2p p2 p2 p2 p2 p2 p2 p2 p2 4 3 2 1 0 − 1 − 2 − 3 − 4 − 3 − 2 − 1 0 1 2 3 4 cos α 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 (Image Credit: [4]) Page 1 of 4 version: 2017/10/05 14:56 M.Eng. René Schwarz (rene-schwarz.com): Memorandum Series Trigonometric Functions (Memorandum № 3) 4 Pythagorean and related trigonometric identities cos2 α + sin2 α = 1 in terms of sin α cos α tan α cot α sec α csc α p (Pythagorean Identity, p tan α 1 sec2 α − 1 1 sin α = sin α 1 − cos2 α p p named after 1 + tan2 α 1 + cot2 α sec α p csc α Pythagoras of Samos p 1 cot α 1 csc2 α − 1 cos α = 1 − sin2 α cos α p p Ὁ Πυθαγόρας ὁ Σάμιος) 2 2 p 1 + tan α 1 + cot α sec α csc α sin α 1 − cos2 α 1 p 1 2 1 tan α = p tan α sec2 α − 1 p 1 + tan α = − 2 α α 2 − cos2 α p1 sin α cos cot csc α 1 1 − sin2 α cos α 1 1 p = sec2 α cot α = p cot α p csc2 α − 1 sin α 1 − cos2 α tan α p sec2 α − 1 1 1 1 p 1 + cot2 α csc α 1 + cot2 α = sec α = p 1 + tan2 α sec α p sin2 α 1 − sin2 α cos α p cot α csc2 α − 1 p 2 1 1 1 + tan2 α sec α = csc α csc α = p 1 + cot2 α p csc α sin α 1 − cos2 α tan α sec2 α − 1 5 Symmetry, shifts and periodicity Symmetry Shifts Periodicity (k 2 Z) − − π − − π · sin( α) = sin α x = 2 α π α α + 2 α + π α + 2π sin(α + k 2π) = sin α − cos(−α) = cos α sin x = cos α sin α cos α sin α sin α cos(α + k · 2π) = cos α cos x = sin α − cos α − sin α − cos α cos α tan(−α) = − tan α tan(α + k · π) = tan α tan x = cot α − tan α − cot α tan α tan α (−α) = − α · cot cot cot x = tan α − cot α − tan α cot α cot α cot(α + k π) = cot α sec(−α) = sec α sec x = csc α − sec α − csc α − sec α sec α sec(α + k · 2π) = sec α csc(−α) = − csc α csc x = sec α csc α sec α − csc α csc α csc(α + k · 2π) = csc α 6 Addition theorems, sum and difference identities of two functions α+β α−β sin(αβ) sin(α β) = sin α cos β cos α sin β sin α + sin β = 2 sin 2 cos 2 tan α tan β = cos α cos β ∓ − α+β α−β sin(αβ) cos(α β) = cos α cos β sin α sin β sin α sin β = 2 cos 2 sin 2 cot α cot β = sin α sin β tan αtan β α+β α−β cos(α−β) tan(α β) = 1∓tan α tan β cos α + cos β = 2 cos 2 cos 2 tan α + cot β = cos α sin β cot α cot β∓1 − − α+β α−β − cos(α+β) cot(α β) = cot βcot α cos α sin β = 2 sin 2 sin 2 cot α tan β = sin α cos β 7 Products of two functions 1 − − 1 − sin α sin β = 2 (cos(α β) cos(α + β)) sin α cos β = 2 (sin(α β) + sin(α + β)) 1 − − 1 − cos α sin β = 2 (sin(α β) sin(α + β)) cos α cos β = 2 (cos(α β) + cos(α + β)) 8 Power reduction formula Sine Cosine n−1 ( ) n−1 X2 − ( ) X2 ( ) 2 n 1 −k n 2 n n odd sinn α = (−1) 2 sin ((n − 2k)α) cosn α = cos ((n − 2k)α) 2n k 2n k k=0 k=0 n −1 n −1 ( ) 2X ( ) ( ) 2X ( ) n 1 n 2 n −k n n 1 n 2 n n even sin α = + (−1)( 2 ) cos ((n − 2k)α) cos α = + cos ((n − 2k)α) 2n n 2n k 2n n 2n k 2 k=0 2 k=0 1 − cos(2α) 3 sin α − sin(3α) cos(4α) − 4 cos(2α) + 3 sin2 α = sin3 α = sin4 α = 2 4 8 1 + cos(2α) cos(3α) + 3 cos α cos(4α) + 4 cos(2α) + 3 cos2 α = cos3 α = cos4 α = 2 4 8 Page 2 of 4 version: 2017/10/05 14:56 M.Eng. René Schwarz (rene-schwarz.com): Memorandum Series Trigonometric Functions (Memorandum № 3) 9 Multiple-angle formulæ Pn ( ) n k n−k π(n−k) tan(nθ)+tan θ sin(nθ) = k cos θ sin θ sin 2 tan((n + 1)θ) = 1−tan(nθ) tan θ k=0 Pn ( ) n k n−k π(n−k) cot(nθ) cot θ−1 cos(nθ) = k cos θ sin θ cos 2 cot((n + 1)θ) = cot(nθ)+cot θ k=0 sin x cos x tan x cot x 2 tan α 2 − 2 1−tan2 α 2 tan α cot2 α−1 x = 2α 2 sin α cos α = 1+tan2 α cos α sin α = 1+tan2 α 1−tan2 α 2 cot α − 3 3 − 3 tan α−tan3 α cot3 α−3 cot α x = 3α 3 sin α 4 sin α 4 cos α 3 cos α 1−3 tan2 α 3 cot2 α−1 3 − 4 − 2 4 tan α−4 tan3 α cot4 α−6 cot2 α+1 x = 4α 8 cos α sin α 4 cos α sin α 8 cos α 8 cos α + 1 1−6 tan2 α+tan4 α 4 cot3 α−4 cot α 10 Half-angle formulæ q q α 1 − α 1−cos α 1−cos α sin α − sin 2 = 2 (1 cos α) tan 2 = 1+cos α = sin α = 1+cos α = csc α cot α q q α 1 α 1+cos α 1+cos α sin α cos 2 = 2 (1 + cos α) cot 2 = 1−cos α = sin α = 1−cos α = csc α + cot α 11 Inverse trigonometric functions 11.1 Definition of the inverse trigonometric functions Trig.