Essential Specialist Mathematics Third Edition With Student Cd-rom

P1: GHM 9780521609999c01.xml CUAU063-EVANS July 23, 2009 21:11 C H A P T E R 1 A toolbox Objectives To revise the properties of sine, cosine and tangent To revise methods for solving right-angled triangles To revise the sine rule and cosine rule To revise basic triangle, parallel lines and circle geometry To revise arithmetic and geometric sequences To revise arithmetic and geometric series To revise infinite geometric series To revise cartesian equations for circles To sketch graphs of ellipses from the general cartesian relation (y − k)2 (x − h)2 + =1 a2 b2 To sketch graphs of hyperbolas from the general cartesian relation (y − k)2 (x − h)2 − =1 2 a b2 To consider asymptotic behaviour of hyperbolas To work with parametric equations for circles, ellipses and hyperbolas The first six sections of this chapter revise areas for which knowledge is required in this course, and which are referred to in the Specialist Mathematics Study Design. The final section introduces cartesian and parametric equations for ellipses and hyperbolas. y 1.1 Circular functions (0, 1) Defining sine, cosine and tangent The unit circle is a circle of radius one with centre at the origin. It is the graph of the relation x 2 + y 2 = 1. (_1, 0) x 0 (0, _1) 1 (1, 0) P1: GHM 9780521609999c01.xml 2 CUAU063-EVANS July 23, 2009 21:11 Essential Specialist Mathematics Sine and cosine may be defined for any angle through the unit circle. For the angle of  ◦ , a point P on the unit circle is defined as illustrated opposite. The angle is measured in an anticlockwise direction from the positive direction of the x axis. cos( ◦ ) is defined as the x-coordinate of the point P and sin( ◦ ) is defined as the y-coordinate of P. A calculator gives approximate values for these coordinates where the angle is given. y P(cos (θ°), sin (θ°)) θ° x 0 y y y (0.8660, 0.5) 30° 0 (_0.1736, 0.9848) (_0.7071, 0.7071) 100° 135° x 0 x 0 1 sin 135◦ = √ ≈ 0.7071 2 −1 cos 135◦ = √ ≈ −0.7071 2 sin 30◦ = 0.5 (exact value) √ 3 ◦ ≈ 0.8660 cos 30 = 2 tan( ◦ ) is defined by tan( ◦ ) = x cos 100◦ ≈ −0.1736 sin 100◦ ≈ 0.9848 sin( ◦ ) . The value of tan( ◦ ) can be illustrated geometrically cos( ◦ ) through the unit circle. By considering similar triangles OPP  and OTT  , it can be seen that i.e. PP  TT  = OT  OP  sin( ◦ ) TT  = = tan( ◦ ) cos( ◦ ) y P T(1, tan (θ°)) O θ° P' T' x sin (θ°) = PP' For a right-angled triangle OBC, a similar triangle OB C  can be constructed that lies in the unit circle.   = sin( ◦ ). By the definition, OC  = cos( ◦ ) and C B The scale factor is the length OB. Hence BC = OB sin( ◦ ) and OC = OB cos( ◦ ). This implies BC = sin( ◦ ) OB and OC = cos( ◦ ) OB B B' 1 O θ° C' C P1: GHM 9780521609999c01.xml CUAU063-EVANS July 23, 2009 21:11 Chapter 1 — A toolbox This gives the ratio definition of sine and cosine for a right-angled triangle. The naming of sides with respect to an angle  ◦ is as shown.   opp opposite ◦ sin  = hyp hypotenuse   adj