By Woerle K.
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Extra resources for 200 Aufgaben aus der Trigonometrie mit Loesungen
Y (mi) 1 P(x, y) Ϫ1 1 x (mi) Ϫ1 FIGURE 1 (a) The dots give the position of markers. (b) An equation of the curve C is 4x4 Ϫ 4x 2 ϩ y 2 ϭ 0. First, the equation does not define y explicitly as a function of x or x as a function of y. 2). Because of this, we cannot make direct use of many of the results for functions developed earlier. Second, the equation does not tell us when the yacht is at a given point (x, y). Third, the equation gives no inkling as to the direction of motion of the yacht. To overcome these drawbacks when we consider the motion of an object in the plane or plane curves that are not graphs of functions, we turn to the following representation.
Use a CAS to approximate the distance traveled by the planet during one complete orbit around the sun. (a) y 105. The Reflective Property of the Parabola The figure shows a parabola with equation y 2 ϭ 4px. The line l is tangent to the parabola at the point P(x 0, y0). y å ∫ å l P(x0, y0) 0 F1 ∫ Tangent line at P P(x0, y0) F2 x ∫ å 0 ƒ F( p, 0) (b) x Show that a ϭ b by establishing the following: 2p y0 a. tan b ϭ b. tan f ϭ y0 x0 Ϫ p 2p c. tan a ϭ y0 Hint: tan a ϭ 108. Reflecting Telescopes The reflective properties of the parabola and the hyperbola are exploited in designing a reflecting telescope.
Ax 2 ϩ Cy 2 ϩ Dx ϩ Ey ϩ F ϭ 0 is an ellipse. 110. The graph of y 4 ϭ 16ax 2, where a Ͼ 0, is a parabola. 111. The ellipse b x ϩ a y ϭ a b , where a Ͼ b Ͼ 0, is contained in the circle x 2 ϩ y 2 ϭ a 2 and contains the circle x 2 ϩ y 2 ϭ b 2. 2 2 2 2 2 114. If A and C have opposite signs, then Ax 2 ϩ Cy 2 ϩ Dx ϩ Ey ϩ F ϭ 0 is a hyperbola. Plane Curves and Parametric Equations Why We Use Parametric Equations Figure 1a gives a bird’s-eye view of a proposed training course for a yacht. In Figure 1b we have introduced an xy-coordinate system in the plane to describe the position of the yacht.