Question

11.2.11 Find an equation for the line tangent to the curve at the point defined by the given value of t. Also, find the value of at this point. x=t-sin ty=1 - 3 cos tt Write the equation of the tangent line. y= x+ (Type exact answers, using a as needed.)

Answer 1

dura Find an equation for the line tangent to the curre at the point defined by the given value of t.. Also find the value of day at this point. x=t- sint, y = 1-3 cost, t= 93 Griven a = 7-sint , y = 1-3 Lest on differentiating with respect to t. dx = 1- Cost dy = zsint - ③ It on dividing by dy/dt = 3 sint = 3tsint, Cesta drygt - i-cost y sint H2 (3) roo Sint=2 sinta Cestha i dy & 3 Cost/2 = 3 ot (4/2) 1-lest=gsin2tl2] Sinth 11 so, slope of tangent - dye It-n e 31 Now, the equation of targent at (x=t-simt, y = 1-3 Cost) ie ( 5 ), 1-34) is yn yn dy(-24) y = (x- ( 17 - 2) 355 29t1 = (6x-26 +353) (8) 1 0 + 2y pl = 63x - 2530 + 9 24 = 663x - 23 ñ + 8 . y = 353n - u30+ y Henve, the equation of the tangent line. y = [33x + [4-35 from ③ dy = 3 ot(t2) Again, differentiate with respect ton. dy = 3d Cot (th). I dt कर dith) da2 : ds | Now a to =3(Cosee? bile)) cast) = -3 ocl (t) = -3 Cosee (12) 2 2 Sof the) z casetony = 2 . 16 (12)