Which of the following vectorvalued functions represent the same graph? a) r(t)=(−3cos⁡t+1)i+(5sin⁡t+2)j+4k b) r(t)=4i+(−3cos⁡t+1)j+(5sin⁡t+2)k c) r(t)=(3cos⁡t−1)i+(−5sin⁡t−2)j+4k d) r(t)=(−3cos⁡2t+1)i+(5sin⁡2t+2)j+4k

Question

Which of the following vectorvalued functions represent the same graph?  a) r(t)=(−3cos⁡t+1)i+(5sin⁡t+2)j+4k  b)  r(t)=4i+(−3cos⁡t+1)j+(5sin⁡t+2)k  c) r(t)=(3cos⁡t−1)i+(−5sin⁡t−2)j+4k  d) r(t)=(−3cos⁡2t+1)i+(5sin⁡2t+2)j+4k

Louise Eldridge · Accepted Answer

Step 1 Since r(t)=ξ+yi+zk a) r(t)=(−3cos⁡t+1)i+(5sin⁡t+2)j+4k Parametric equations, x=−3cos⁡t+1 y=5sin⁡t+2  z=4 Then, cos⁡t=1−x3 sin⁡t=y−25 And cos2t+sin2t=1 (1−x3)2+(y−25)2=1 (y−25)2=1−1−2x+x29 (y−25)2=8+2x−x29 y=538+2x−x2+2 b) r(t)=4i+(−3cos⁡t+1)j+(5sin⁡t+2)k Parametric equations, x=4 y=−3cos⁡t+1 z=5sin⁡t+2 Hence, cos⁡t=1−y3 sin⁡t=z−25 And cos2t+sin2t=1 (1−y3)2+(z−25)2=1 (1−y3)2=1−4−4z+z225 (1−y3)2=25+4z−z225 y=1−3521+4z−z

Which of the following vectorvalued functions represent the same graph? a)

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