Trigonometry summary sheet, Summaries of Trigonometry

Download Trigonometry summary sheet and more Summaries Trigonometry in PDF only on Docsity! QUADRANTS & SIGNS OF FUNCTIONS ___________________________________________________________________________ RIGHT-ANGLE TRIANGLE RELATIONSHIPS a = opposite side b = adjacent side c = hypotenuse Pythagorean theorem: ___________________________________________________________________________ FUNDAMENTAL IDENTITIES 1. 2. 3. 4. 5. 6. 7. 8a. 8b. 9a. 9b. 10. OPPOSITE-ANGLE IDENTITIES 1. 2. 3. 4. 5. 6. ADDITION LAWS 1. 2. 3. DOUBLE-ANGLE IDENTITIES 1a. 1b. 1c. 1d. 1e. 2. 3. HALF-ANGLE IDENTITIES 1. 2. 3a. 3b. PRODUCT IDENTITIES 1. 2. 3. 4. 5. SUM IDENTITIES 1. 2. 3. 4. ___________________________________________________________________________ REDUCTION IDENTITY 1. , where is chosen so that and ___________________________________________________________________________ PLANE TRIANGLE RELATIONSHIPS Law of sines: Law of cosines: , Law of tangents: ___________________________________________________________________________ INVERSE TRIGONOMETRIC FUNCTIONS PRINCIPAL VALUES FOR INVERSE TRIGONOMETRIC FUNCTIONS INVERSE IDENTITIES (ASSUMING PRINCIPAL VALUES ARE USED) 1. 7. 2. 8. 3. 9. 4. 10. 5. 11. 6. 12. ___________________________________________________________________________ COMPLEX IDENTITIES Euler Identity: ___________________________________________________________________________ QUADRATIC FORMULA Solution to : ___________________________________________________________________________ EXACT VALUES III III IV All positive sin & csc pos. others neg. cos & sec pos. others neg. tan & cot pos. others neg. θ 90° - θ a b c 90° θsin side opposite θ hypotenuse ----------------------------------- a c --= = θcos side adjacent to θ hypotenuse ----------------------------------------- b c --= = θtan side opposite θ side adjacent to θ ----------------------------------------- a b --= = c2 a2 b2+= xtan xsin xcos -----------= xsec 1 xcos -----------= xcsc 1 xsin ----------= xcot 1 xtan ---------- xcos xsin -----------= = x2cos x2sin+ 1= 1 x2tan+ x2sec= x2cot 1+ x2csc= π 2⁄ x–( )cos xsin= π 2⁄ x+( )cos xsin–= π 2⁄ x–( )sin xcos= π 2⁄ x+( )sin xcos= π 2⁄ x–( )tan xcot= x–( )cos xcos= x–( )sin xsin–= x–( )tan xtan–= x–( )sec xsec= x–( )csc xcsc–= x–( )cot xcot–= x y±( )cos x ycoscos x ysinsin+−= x y±( )sin x ycossin x ysincos±= x y±( )tan xtan ytan± 1 x ytantan+− -------------------------------= 2xcos x2cos x2sin–= 2xcos 2 x2cos 1–= 2xcos 1 2 x2sin–= x2sin 1 2 -- 1 2xcos–( )= x2cos 1 2 -- 1 2xcos+( )= 2xsin 2 x xcossin= 2xtan 2 xtan 1 x2tan– ---------------------= x 2 --cos 1 xcos+ 2 --------------------±= + if x/2 is in quadrant I or IV - if x/2 is in quadrant II or III   x 2 --sin 1 xcos– 2 --------------------±= + if x/2 is in quadrant I or II - if x/2 is in quadrant III or IV   x 2 --tan 1 xcos– 1 xcos+ --------------------±= + if x/2 is in quadrant I or III - if x/2 is in quadrant II or IV   x 2 --tan 1 xcos– xsin -------------------- xsin 1 xcos+ -------------------- xcsc xcot–= = = 2 xcos ycos x y–( )cos x y+( )cos+= 2 xsin ysin x y–( )cos x y+( )cos–= 2 xsin ycos x y+( )sin x y–( )sin+= 2 x ysincos x y+( )sin x y–( )sin–= mxcos nxcos m n+( )xcos m n–( )xcos+= xcos ycos+ 2 x y+ 2 -----------   cos x y– 2 ----------   cos= xcos ycos– 2 x y+ 2 -----------   sin x y– 2 ----------   sin–= xsin ysin+ 2 x y+ 2 -----------   sin x y– 2 ----------   cos= xsin ysin– 2 x y– 2 ----------   sin x y+ 2 -----------   cos= a xsin b xcos+ a2 b2+ x y+( )sin= y ycos a a2 b2+ -------------------= ysin b a2 b2+ -------------------= Function Domain Range Quadrants I and II I and IV all reals I and IV or , I and II or , I and IV all reals I and II Principal values for Principal values for Angle (deg) Angle (rad) cos(x) sin(x) tan(x) 1 0 0 1 0 1 -1 0 0 0 -1 A B C a b c a Asin ----------- b Bsin ----------- c Csin -----------= = c2 a2 b2 2ab Ccos–+= C a2 b2 c2–+ 2ab --------------------------   acos= a b+ a b– ----------- 1 2 -- A B+( )tan 1 2 -- A B–( )tan ------------------------------= y x1–cos= 1– x 1≤ ≤ 0 y π≤ ≤ y x1–sin= 1– x 1≤ ≤ π 2⁄– y π 2⁄≤ ≤ y x1–tan= π 2⁄– y π 2⁄< < y x1–sec= x 1≥ x 1–≤ 0 y π≤ ≤ y π 2⁄≠ y x1–csc= x 1≥ x 1–≤ π 2⁄– y π 2⁄≤ ≤ y 0≠ y x1–cot= 0 y π< < x 0≥ x 0< 0 x1–sin π 2⁄≤ ≤ π 2⁄– x1–sin 0<≤ 0 x1–cos π 2⁄≤ ≤ π 2⁄ x1–cos π≤< 0 x1–tan π 2⁄<≤ π 2⁄– x1–tan 0< < 0 x1–cot π 2⁄≤< π 2⁄ x1–cot π< < 0 x1–sec π 2⁄<≤ π 2⁄ x1–sec π≤< 0 x1–csc π 2⁄≤< π 2⁄– x1–csc 0<≤ x1–sin x1–cos+ π 2⁄= x–( )1–sin x1–sin–= x1–tan x1–cot+ π 2⁄= x–( )1–cos π x1–cos–= x1–sec x1–csc+ π 2⁄= x–( )1–tan x1–tan–= x1–csc 1 x⁄( )1–sin= x–( )1–cot π x1–cot–= x1–sec 1 x⁄( )1–cos= x–( )1–sec π x1–sec–= x1–cot 1 x⁄( )1–tan= x–( )1–csc x1–csc–= wcos eiw e iw–+ 2 --------------------= i wsin eiw e iw–– 2 --------------------= eiw wcos i wsin+= ax2 bx c+ + 0= x b– b2 4ac–± 2a -------------------------------------= 0° 0 30° π 6⁄ 3 2⁄ 1 2⁄ 3 3⁄ 45° π 4⁄ 2 2⁄ 2 2⁄ 60° π 3⁄ 1 2⁄ 3 2⁄ 3 90° π 2⁄ ∞± 180° π 270° 3π 2⁄ ∞± TRIGONOMETRY REVIEW SHEET ___________________________________________________________________________