38 Chapt. 4 Two- and Three-Dimensional Kinematics 004 qmult 00470 1 1 5 easy memory: small angle approximations 23. For small angles θ measured in radians and with increasing accuracy as θ goes to zero (where the formulas are in fact exact), one has the small angle approximations: a) sin θ ≈ cos θ ≈ 1 − 1 2 θ 2 . b) cos θ ≈ tan θ ≈ 1 − 1 2 θ 2 . c) sin θ ≈ cos θ ≈ θ . d) cos θ ≈ tan θ ≈ θ . e) sin θ ≈ tan θ ≈ θ . 004 qmult 00500 1 1 5 easy memory: polar coordinates 24. In 2-dimensional Cartesian coordinates, a displacement vector v r is given by v r = ( x,y ) = x ˆ x + y ˆ y , where the unit vectors ˆ x and ˆ y are constants. In polar coordinates, v r = ( r,θ ) = r ˆ r , where the unit vector ˆ r = cos θ ˆ x + sin θ ˆ y . The polar coordinates are obtained from the Cartesian ones by the formulae r = r x 2 + y 2 and θ = tan − 1 p y x P . In calculational work one must be aware that a negative argument of tan
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