tan t = y x, x Z 0 sin u 2 = ; 1 - cos u cos u 2 = ; 1 + cos u tan u 2 = 1 - cos u cos a cos b = 1 2 sin a cos b = 1 2

Transcription

1 TRIGONOMETRIC FUNCTIONS Let t be a real number and let P =, be the point on the unit circle that corresponds to t. sin t = cos t = tan t =, Z 0 csc t =, Z 0 sec t =, Z 0 cot t =. Z 0 P (, ) t s t units t units 0 t radians t t radians 0 t units TRIGONOMETRIC IDENTITIES Fundamental Identities tan u = sin u cos u cot u = cos u csc u = sec u = sin u sin u + cos u = tan u + = sec u cot u + = csc u sin u cos u cot u = tan u Half-Angle Formulas sin u = ; - cos u A cos u = ; + cos u A tan u = - cos u sin u P (, ) s t units Double-Angle Formulas sin u = sin u cos u cos u = cos u - sin u cos u = cos u - cos u = - sin u tan u tan u = - tan u Even-Odd Identities sin -u = -sin u csc -u = -csc u cos -u = cos u sec -u = sec u tan -u = -tan u cot -u = -cot u Product-to-Sum Formulas sin a sin b = Ccos a - b - cos a + bd cos a cos b = Ccos a - b + cos a + bd sin a cos b = Csin a + b + sin a - bd Sum and Difference Formulas sin a + b = sin a cos b + cos a sin b sin a - b = sin a cos b - cos a sin b cos a + b = cos a cos b - sin a sin b cos a - b = cos a cos b + sin a sin b tan a + b = tan a - b = tan a + tan b - tan a tan b tan a - tan b + tan a tan b Sum-to-Product Formulas sin a + sin b = sin a + b cos a - b sin a - sin b = sin a - b cos a + b cos a + cos b = cos a + b cos a - b cos a - cos b = - sin a + b sin a - b SOLVING TRIANGLES c a b Law of Sines sin a a = sin b b = sin g c Law of Cosines a = b + c - bc cos a b = a + c - ac cos b c = a + b - ab cos g

2 FORMULAS/EQUATIONS Distance Formula If P =, and P =,, the distance from to is Standard Equation of a Circle dp, P = The standard equation of a circle of radius r with center at (h, k) is - h + ( - k) = r Slope Formula The slope m of the line containing the points P =, and P =, is m = - - if Z m is undefinedif = Point Slope The equation of a line with slope m containing the point, is Equation of a Line Slope Intercept - = m - The equation of a line with slope m and -intercept b is Equation of a Line Quadratic Formula = m + b The solutions of the equation a + b + c = 0, a Z 0, are = -b ; b - 4ac a If b - 4ac 7 0, there are two unequal real solutions. If b - 4ac = 0, there is a repeated real solution. If b - 4ac 6 0, there are two comple solutions that are not real. P P GEOMETRY FORMULAS Circle r r = Radius, A = Area, C = Circumference A = pr C = pr Triangle b h b = Base, h = Altitude Height, A = area A = bh Rectangle l w l=length, w=width, A=area, P=perimeter A=lw P=l+w Rectangular Bo l h w l=length, w=width, h=height, V=Volume, S=Surface area V=lwh S=lw+lh+wh Sphere r r=radius, V=Volume, S=Surface area V = 4 pr S = 4pr Right Circular Clinder r h r=radius, h=height, V=Volume, S=Surface area V=pr h S=pr +prh

3 LIBRARY OF FUNCTIONS Identit Function Square Function Cube Function f = f = f = (, 4) 4 (, 4) 4 (, ) (, ) (0, 0) 4 (, ) (0, 0) (, ) 4 4 (, ) 4 (, ) (0, 0) 4 Square Root Function Reciprocal Function Cube Root Function f = f = f = (, ) (0, 0) (4, ) 5 (, ) (, ) (, ) (, ) 8 (, ) (, ) (, ) (, 8 ) (0, 0) (, ) Absolute Value Function Eponential Function Natural Logarithm Function f = ƒƒ f = e f = ln (, 7.9) (,.0) (, ) (, ) 6 (, ) (, ) (, 0.69) (0, 0) (,.7) (, 0.7) (0, ) (, 0), 0.69 ( ) Sine Function Cosine Function Tangent Function f = sin f = cos f = tan Cosecant Function Secant Function Cotangent Function f = csc f = sec f = cot 5

4 Prepare for Class Read the Book Feature Description Benefit Page Chapter Opening Each chapter begins with a current The Article describes a real situation. The 47 Article & Project article and ends with a related project. Project lets ou appl what ou learned to solve a related problem. Ever section begins with Learning Each section begins with a list of objectives. These focus our studing b 48 Objectives Objective numbers appear in the margin emphasizing what s most important where the objective is covered. and where to find it. Ever chapter begins with. Most sections contain Preparing for Sections begin with a list of ke Ever forget what ou ve learned? This feature 48 This Section concepts to review, with page numbers. highlights previousl learned material to be used in this section. Review it, and ou ll alwas be prepared to move forward. Now Work the Special problems that support the Not sure ou need the Preparing for This 48 & 5 Are You Preparing for This Section feature. Section review? Work the Are You Prepared? Prepared? problems. If ou get one wrong, ou ll know Problems eactl what ou need to review and where to review it! Calculus Icon These appear net to information Pa attention if ou spend etra 5 essential for the stud of calculus. time now, ou ll do better later! Now Work These follow most eamples, and We learn best b doing. You ll solidif 50 Problems direct ou to a related eercise. our understanding of eamples if ou tr a similar problem right awa, to be sure ou understand what ou've just read. WARNING Warnings are provided in the tet. These point out common mistakes 04 and help ou to avoid them. Seeing the These features suggest graphing You will obtain a deeper and more 5 & 65 Concept & utilit activities. intuitive understanding of theorems and Eplorations definitions. In Words These provide alternative descriptions Does math ever look foreign to ou? 57 of select definitions and theorems. This feature translates math into plain English. Step-b-Step, Eamples contain detailed, Work the eamples on our own, uncovering Annotated intermediate steps. Man include the solution line-b-line as ou go. Each line Eamples additional annotations. will verif our work. For additional help, 65 consult the blue annotations, which provide the reason wh the statement to the left is valid. Side-b-Side Eamples that can be solved Seeing the solution side-b-side Algebraic and both algebraicall and graphicall will help ou to decide the more Graphing are shown side-b-side. efficient wa to solve a problem. Solutions

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