WORLD'S #1 AC ADEM IC OUT LINE BarC h a r ts Inc.'" Trigonometry (trig) means measurcment of triangles; it is usually studicd as measurcmcnts of sides and angles of triangl es and as points on a unit circle; this study guide is basically separated into these two main sections: trig with triangles and trig with a unit circle; in trigonometry, the measures of angles are usually represe nted by letters Crom the G reek alphabet; the Greek letters e, fl.• v and ~ will be used throughout this study guide to repre sent angle mcasurcs TRIG WITH TRIANGLES ._ - A Right Tr ian gle I A ri ght tria ngle is a triangle w ith exactl y onc 90° (right) angle T he hypotenuse is the longest si de of a right triangle and is always located opposi te the right (90°) angle The two shorter sides o f a right tri angle are both cal led legs The Pythagorea n T heore m (Ieg' + leg' = hypotenuse' or a' + b' = c' where a a nd b a rc leg lengths and c is th e hy potenuse length) may be used to f ind the length of a ny thi rd si de ofa ri ght t ri ang le w hen any two side lengths are known a Whe n the two leg le ng ths a2 + bl = c2 a rc know n squa re the le ngth of eac h leg, a dd 92 + 16 = c2 these two square s togethe r 81 + 256 = CZ a nd squa re root th e res ult 337 = CZ ing sum; for example: -,/337 = c The o pposite leg ofa rig ht triangle is the leg w hic h does not touch the vertex of the angle th a t is na med in the trig func tion x Th e adj a ce nt leg of a r ig ht tria ng le is the leg A " " - - - - - -, , which does touch the vertex of th e angle that is named in the trig func tion; for examp le: When evaluating the trig functio ns for angle A in this right triangle, leg y is the oppos ite leg z y for angle A becallse it does not tOllch point A; however, leg x is the adjacent leg for ang le A because it does touch point A; the hypotenuse B is side~ In the same righ t triangle, leg x is the oppos ite leg for angle R because it does not to uch point B; howeve r, Ieg y is the adja cent leg for angle B because it does to uch point Ii (NO TICE: Tire leg tlrut is tire opposite leg f or allgle A is tir e sallie leg tllllt is tir e adja cellt leg for WIgle B, (Illd tire leg that is tire adj([cellt legfilr allgle A is tI,e S([lIIe leg that is tire opposite leg for (/Ilg le H; tire Irypotelluw is 1I('I'er ctllISitiereti as tir e opposite side nor as tire atij acent side because it is /l ot 1I le/:J Since trig functions are rat ios, a nd rat ios can be writ ten as dec ima l numbers, trig functions are either converted to decim a l nu mbers or left as radical expressions in lowest terms (for examp le, '1 : or 866 ); for examp le, in the right triangles above, if: z = 9,/ = 7, aZ + b2 = 82 + b2 = 64 + b2 = bZ = bZ = leg = c hypotenuse adjacent leg cosme e = cos = ~ hypotenuse opposite leg tangent e = tall e = adjacent leg _ x 4;/2 z A BC (as a c b indicated in L'>ABC above in the law of sill ex sill ~ sill v cosines): ii When to apply the law or sines: The law of sines may be used either when one side length and two angle measures are known (SAA, that is, one of the angles must be opposite the side) or when two s ide lengths and one angle measure are known (SSA, that is, the angle tllust be opposite one of the two sides) iii Caution When using the law of sines, occasionally there will be no solution; this is because not all combinations of angle measures and side lengths actually form triangle s: remember that the third side of any triangle must have a length longer than thc differ ence of the other two sides and shorter than the sum ofthesc other two sides sin \ J 14 73" is the angle of elevation a A a C B c Law of Cosines The law of cosines states that in a triangle ABC: a' = b' + c2 - 2bc cos ex b = a' + c' - 2ac cos ~ c' = a' + b2 - 2ab cos v definitions for right triangles can be applied as discussed (NOTE: Another option/or solving aClite triangles is to leave the triangles as they are (aclIte) and to apply the law of cosines or the law or sines, both of which are dis clIssed at the top o/the next column) b Obtuse Trianglcs Any ohtuse triangle (triangle with exactly B one obtuse angle) can be converted into a right triangle by constructing a line segment from one of the vertices and perpendicular to the line containing the side opposite the vertex; for example, in L'>ABC, LC is obtuse; extend side AC, then draw s o perpendicular to the A ""' -:!' - - - extension; the result is right L'>ABD C Then the trig function definitions for right triangles can be applied as discussed above (NOTE: Another option fiJI' solving obtuse triangles is to leave the triangles as they are (obtuse) and to apply either the law o.fcosines or the law ofsines, hoth o.f which are discussed at the top o/the next column) iv For example; if LA = 40° then LA = 7t(40) = 7t(2) 180 = I The domain of the sine function is the set orrcal numbers; the range is the set of real numbers between -I and I inclu sivciy; i.c -I ~ Y ~ I ii, Both the domain and the range of the cosine function are the same as the domain and the range ()fthe sine func tion 27t radians c See the radians and degrees chart under the topic of Unit Circle for the Measllrements ojSpecial Angles • Generated Angles I A generated angle (another ty pe of angle o fte n used in trigonometry) is a central angle with the vertex placed at the origin of the coordinate plane, and one of the two sides placed and kept on th e positive x-axis, while the second side is rotated in either a clockwise or counterclockwise direction a The side that docs not rotate is called the initial side b The side that does rotate is called the terminal side c Negative angles are formed when the terminal side rotates clockwise d Positive angles are f()I"Jne d when the terminal side rotates counte rclockwise x-axis iii The domain of the tangent function is the set of all real numbers except those va lues where the function is undefined and goes off asymptoti.:ally, such as ± ff/2, ± 3n/2 , The range is the set of all real num be rs: the dashed lincs arc the vcrti cal asymptotes initial side D Unit Circle I The unit circle is a circle whose center is the origin (0,0) of the rectangular coordi y= nate plane and whose radius is equal to exactly one unit (radius = I and diameter = 2) The equation of the unit circle is x' + y'= I A point, P, is on the unit circle if and only if the di stance from the center of the circle to the point is equal to the radius of exactly one unit The unit circle is sym metric with respect to the y-axis x-axis, the y-axis, and the origin; therefore if point P = (a ,b) is on the unit circ le, then these points arc also on the unit circle (-a,b) (-a ,-b) and (a,-b); for example: I{{II t - :L\is x-axis II Periods of the functions a A function , I, is periodic if there is a pos itive number VI such that I (t+ v)=/(t) for all tin the domain of the function; this lTlay also be stated using x in placc of the t vulue T he smallest value of v is I:alled the peri od of the function; that is, the smallest value at whkh a function begin s to repeat its range values, and thus n:pcat it s graphing pattern, is the pe~iod of the function b The period orthe sine fun e tion,/(t) =sill t, is 2n: because sill(t + 1t)- sill t c The pe r iod of the cosine function, I (t) =C()st is also 2n: bccall s ~ cos (t+2n:)=co.,· t d The period orthe tangent function , /(t)= tllll t, is n: because' IOIl(t-+1t)= /c1ll t (NOTE: These period,· e(lll be observed ill th" gml'lis o/ Ihejilllctiulls liS illdiwled "h(ln') e The period ofa function,f(t)=sill Bt, is 2n/ 13; the elTect o rthe value off] is that it stretches the graph out hori zo ntally when U< I3< and shrinks the graph horizontally when B> I The distance between any two points on the rectangular coordinate plane may bc fo und by using the formula; ./(x t - x2)2 + (Yt - Yl)2 where the points are (X t , Yt) and (xl> Y2)· The length of an arc of the unit circl e is based on the ci rcumference, n:d = n:2 = 2n:; because, d=2 Points on the Unit Circle a Points can be labeled using the appropriate order pair, (x,y) b Poin ts can also be labeled using the circular arc length determined by the generated angle whose terminal side contains the point, (x,y); for example: Remember the radius of a unit circle is one; r = y c Constructing a right triangle by drawing a perpendicular to the x-axis, and determining the side lengths of the triangle results in the following unit circle trig function definitions Unit circle trig function definitions (sec the diagram above): Where t = radians sill t = sill a = y COst=cosa= x a = degrees ~ x' esc t= csca = I sill and esc; esc a = I/\"ill a cos and sec: sec a = JleGs a lall and col: col a = ' / rtll/ a Y sec t= sec a Frequently used angles and trig functions are indi cated in the following chart a =dl'gn'l' a t := radi'lIls 40 JS ~) 111ll'rc 13 = = sill 2\: \\here 13 = =.!x x cott=cota=y ;YoFO =unddilll'll ~) I ~l I~~ I~ \1\ \.10 i!! ::.rr Ilff 10U -ff-+~-t~ i!! ~ n 4-~+ -~~~~~ ~ \-' ! ·1 t T · d Ian ,\·ill \; (NOTE: The 1"('(1 sectioll orthe graph illl/icall's olle p eriod o/".,· =s ill.r: alld Ihl' hllle section is aile p eriod o(v=sill 2.\) 12 Amplitude a The amplitude ofa trig function y=/; they may only be s impl if ied; for example: (CO.\ }i +\ /11 \ ) I (l-t'C _\ ) B Trig eq uatio ns contain tri g functions and an equals sign ; they may be solved to fi nd the val ues that make them true: algebraic techniques, such as li,ctoring, may be used to solve trig equations: for example: caSrSil7t = Sill(r + t) - Sill( r- t ) cosscast= cos(s + t) + cas(r - t} Sills sillt = cos( , - t) - ('os (s + t) cos2t - cos t- = () can be factored as (cos t - 2)(cost + J) = then cus t - = or cos t+ I = () cos t = or cos t = - I so in the interval () S t:::; 21t t h;L~ IlO vallie or t = 1t t) cos ( s; t ) ( s +t (S-t) ) Sill Sillr + sillt = 2sill ( s; = "~cas SlI1s-SlIIt C Tri g identities arc true for a ll real numbers in the domain ; they may be proven or cass +c _=2c, Ifl varificd ; methods ofpmving or verifying identities include working the lell side of the equation only until it is identical to the right side; working the right side until it is identical to the left side; or, working both sides until they are identical D F unda mental Trig Identities a nd Formulas Most of these should be memori zed because they arc used frequently ~~ ~ cass- cast = -2.1' CREDITS Author: Dr S B Kizli k Artwork: Michael D Adam Lay out: Michael D Adam free & nundi~ re ads of titles at qUlc uay.com NOT E TO STUDENT DISCLAIMER This gu ide is inte nded only for informati onal pur poses , and is no t mcant 10 be i.I subs titu te for professional s pOrl S instructio n Due to its l~o nJ c n scd formal Ihi:- gu ide ca nn ot cover eve ry aspec t of th is spor t Nei t he r BarC rts, its writers dc!Signcrs nor ed iting sta rr arc in any way re sponsi bl e or liable for the usc or mislise orlhe information co ntain ed in thi s gui d e All rights reserved No part o f this publ ication m ay be re produc ed or trans m itt ed in any form, or by any means, e lectron ic or mechanical in c lud ing photocopy, reco rd ing, or any information stora gc and retri eva l sy st em , witho ut w ritt en per m iss ion fro m the publi :':i hcr