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Exercise 1.15: The sine of an angle (specified
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in radians) can be computed by making use of the approximation
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sin
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x
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≈
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x
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if
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x
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is sufficiently small, and the trigonometric
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identity
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sin
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x
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=
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3
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sin
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x
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3
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−
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4
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sin
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3
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x
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3
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to reduce the size of the argument of sin. (For purposes of this
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exercise an angle is considered “sufficiently small” if its magnitude is not
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greater than 0.1 radians.) These ideas are incorporated in the following
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procedures:
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(define (cube x) (* x x x))
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(define (p x) (- (* 3 x) (* 4 (cube x))))
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(define (sine angle)
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(if (not (> (abs angle) 0.1))
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angle
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(p (sine (/ angle 3.0)))))
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How many times is the procedure p applied when (sine 12.15) is
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evaluated?
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What is the order of growth in space and number of steps (as a function of
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a
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) used by the process generated by the sine procedure when
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(sine a) is evaluated?
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