Parametric Equations for Circles and Ellipses

Transcription

1 Lesson 5-8 Parametric Equations for Circles and Ellipses BIG IDEA Parametric equations use separate functions to defi ne coordinates and and to produce graphs Vocabular parameter parametric equations equation for the unit circle in standard form A circle cannot be the graph of a function with equation = f() because there eist man pairs of points with the same first coordinate However, ou can write each coordinate as a function of a third variable We call the variable t A variable that determines other variables is called a parameter When the coordinates of points on a curve (or line) are each epressed with an equation written in terms of a parameter, the equations are called parametric equations Activit 1 Set our graphing utilit to degree mode Step 1 Find out how ou can enter parametric equations into our technolog Mental Math Tell whether the number is positive, zero, or negative a sin 1000º b sin 1000º c cos 73º - sin 73º d 10 sin(10π) Enter t = cos t and t = sin t Step Choose a suitable window and graph the two equations Describe what happens Step 3 If possible, animate the graph and run our animation from 0º to 1080º or from 0 to 6π The equations in Activit 1 are parametric equations for the unit circle This is because an point P on the unit circle can be considered as the image of (1, 0) under a rotation of magnitude θ, and b definition, R θ (1, 0) = (cos θ, sin θ) In this case, θ is the parameter Recall the Pthagorean Identit cos θ + sin θ = 1 Substituting for cos θ and for sin θ, we get + = 1, the equation for the unit circle in standard form Activit Set our graphing utilit to degree mode and choose parametric for graph tpe Step 1 Graph the equations { = = cos sin t t and { = = 3 3 cos sin t t, both for 0º t 360º What is the effect of the or the 3 on the graph? (continued on net page) Parametric Equations for Circles and Ellipses 331

2 Chapter 5 Step Generalize our observations in Step 1 to describe the graph of { = = r r cos sin t t for 0º t 360º Activit suggests that the image of the unit circle under the size change with center (0, 0) and magnitude r is given b the parametric equations { = r cos t This result can be proved using the Pthagorean = r sin t Identit Theorem (Parametric Equation for a Circle) The circle with center (0, 0) and radius r has parametric equations { = r cos t, 0º t 360º or 0 t π = r sin t Proof Rewrite the parametric equations as { _ r = cos t _ r = sin t We know cos t + sin t = 1, because of the Pthagorean Identit Substitute _ r for cos t and _ r for sin t ( _ r ) + ( _ r ) = 1 Use the Power of a Quotient Propert _ ( r ) + _ ( ) = 1 r Multipl both sides of the equation b r + = r This is an equation for the circle centered at the origin with radius r Notice that multipling cos t and sin t b r makes them r times as large; however, this transformation is equivalent to replacing with _ r and with _ r This substitution is eactl what the Graph Scale-Change Theorem states: the unit circle + = 1 is transformed b a size change of magnitude r QY1 Scale Changes and Parametric Equations B multipling - and -coordinates b constants, ou produce a scalechange image When the constants are not equal, the image of the unit circle under such a transformation is not a circle, but an ellipse QY1 a + = 64 is the image of the unit circle under a size change of what magnitude? b Write parametric equations for this circle GUIDED Eample 1 a Graph the ellipse { = cos t, 0º t 360º = 5 sin t b Write an equation in rectangular coordinates for the ellipse Solution a Make a table of values for 0º t 90º Also include t = 180º and t = 70º in the table Some values have been fi lled in for ou Plot the points on a rectangular grid Use smmetries over the aes to complete a sketch of the ellipse t = = cos t 5 sin t 0º º? 50 60º?? 90º 0? 180º?? 70º? Trigonometr

3 b cos t + sin t = 1 Pthagorean Identit? +? = 1 Substitute _ for cos t and? for sin t Activit 3? +? = 1 Appl the Power of a Quotient Propert Step 1 Graph the ellipse { = = 5 cos sin t t, 0º t 360º, from Eample 1 Step On the same grid, graph { = cos t + 4 = 5 sin t - 3, for 0º t 360º Describe the differences between the two graphs The ellipse of Step 1 of Activit 3 can be mapped onto the ellipse of Step b the translation (, ) ( + 4, - 3) This result suggests the following theorem Theorem (Parametric Equation for a Circle with Center (h, k)) The circle with center (h, k) and radius r has parametric equations { = h + r cos t, 0º t 360º or 0 t π = k + r sin t Proof From the parametric equations, { - - h k = = r r cos sin t t Thus, ( - h) + ( - k) = r cos t + r sin t = r (cos t + sin t) = r 1 = r Eample Write parametric equations for the circle with center ( 4, 5) and radius GUIDED Solution The circle with radius and center ( 4, 5) is the image of the graph of { = cos t fi rst under S: (, ) (, ) and then the translation T: (, ) ( - 4, + 5) Under S, the equations for the image of the circle are { =? =? To move the center to ( 4, 5), add? to the -coordinates and? to the -coordinates Therefore, parametric equations for this circle are { =? + cos t = 5 +? Check Graph the parametric equations using a graphing utilit Parametric Equations for Circles and Ellipses 333

4 Chapter 5 QY Questions COVERING THE IDEAS In 1 and, write parametric equations for the circle described 1 center (3, ) and radius 5 ( - 8) + ( + 4) = 9 3 Write an equation for the ellipse { = 4 cos t in rectangular form 4 Write equations of the circle graphed at the right in standard rectangular and parametric form 5 Write an equation of the circle { = cos t, 0 t π in = + 5 sin t standard rectangular form 6 Write parametric equations for the lower half of the circle with center ( 4, 3) and radius 6 7 Write parametric equations for the circles at the right 8 a Graph { = cos t, 360º t 70º b Compare this with the graph of { = cos t, 0º t 360º 9 As t increases from 0º to 360º, what happens to the corresponding point on the graph of { = 8 cos t = 7 sin t? 10 The unit circle is translated 6 units to the left and 3 units up a Write an equation in rectangular form for the transformed circle b Write parametric equations for the original circle and its image 6 4 QY What is an equation in standard form for the circle of Eample? APPLYING THE MATHEMATICS 11 A circle with center at (8, 3) has a radius of 05 This circle is the image of the unit circle under what transformation? 1 Let S(, ) = (4, 4) and T(, ) = ( -, + 5) Find equations for the image of the graph of { = cos t under a S T b T S 13 a Graph the parametric equations { = 5 cos t = 3 sin t b Describe the shape of the graph c What transformation has been applied to the unit circle in the horizontal direction? In the vertical direction? d Describe how the graph differs from the graph of { = 5 cos t = 5 sin t e Write an equation in standard form that has the same graph as the equation in Part d 334 Trigonometr

5 14 The unit circle is transformed with the mapping S: (, ) (9, 9) Find a mapping that will transform the image back to the unit circle 15 Consider the sets of parametric equations below Compare and contrast the curves the trace out, and how those curves are traced (Man graphing utilities have an animation mode that shows a point moving along a parametric curve) a { = cos t REVIEW b { = sin t = cos t 16 Consider 8 cos θ = 3 - cos θ (Lesson 5-7) a Find all solutions in the interval 0 θ π b Find the general solution c { = cos(t) = sin(t) 17 How man solutions are there to the equation 6 sin(3πt) = when 0 < t < 8? (Lesson 5-7) 18 A hill slopes upward at an angle of 6º with the horizontal A tree grows verticall on the hill When the angle of elevation of the Sun is 4º, the tree casts a shadow 41 m long If the shadow is entirel on the hill, how tall is the tree? (Lesson 5-5) 19 Consider the function with equation = 3 cos ( _ - π 6 ) + 7 Give the amplitude, period, vertical shift, and phase shift of the function (Lesson 4-9) 0 For the function with equation = tan ( + π_ 4 ), determine the (Lesson 4-8) a domain b range c period 1 Given f() = and g() = 1 -, let h() = f(g()) (Lesson 3-8) a Write an epression for h() b State the domain and range of h m EXPLORATION Use parametric equations to construct the picture at the right QY ANSWERS 1 a 8 b { = 8 cos t = 8 sin t ( + 4) + ( 5) = 4 Parametric Equations for Circles and Ellipses 335