Engineering Calculus Notes 169

# Engineering - e a bi = e a(cos b i sin t prove the identities cosh t = cos it sinh t = − i sin it 7(a A wheel of radius 1 in the plane rotating

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2.2. PARAMETRIZED CURVES 157 (b) The circle of radius 1, centered at (1 , 1 , 1), and lying in the plane x + y + z = 3. (c) A curve lying on the cone z = r x 2 + y 2 which rotates about the z -axis while rising in such a way that in one rotation it rises 2 units. ( Hint: Think cylindrical.) (d) The great circle 10 on the sphere of radius 1 about the origin which goes through the points (1 , 0 , 0) and ( 1 3 , 1 3 , 1 3 ). Theory problems: 5. Using the deFnition of the hyperbolic cosine and sine (Equation ( 2.19 )), prove that they satisfy the identities: (a) cosh 2 t sinh 2 t = 1 . (b) cosh 2 t = 1 2 (1 + cosh 2 t ) (c) sinh 2 t = 1 2 (cosh 2 t 1) Challenge problem: 6. Using Euler’s formula
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Unformatted text preview: e a + bi = e a (cos b + i sin t ) prove the identities cosh t = cos it sinh t = − i sin it. 7. (a) A wheel of radius 1 in the plane, rotating counterclockwise with angular velocity ω 1 rotations per second, is attached to the end of a stick of length 3 whose other end is Fxed at the origin, and which itself is rotating counterclockwise with angular velocity ω 2 rotations per second. Parametrize the motion of a point on the rim of the wheel. 10 A great circle on a sphere is a circle whose center is the center of the sphere....
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## This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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