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**Unformatted text preview: **this case reduces to 6 cos(φ) = 0. 3
5 11.1 Applications of Sinusoids 755 It is possible, though beyond the scope of this course, to model the eﬀects of friction and other
external forces acting on the system.15 While we may not have the Physics and Calculus background
to derive equations of motion for these scenarios, we can certainly analyze them. We examine three
cases in the following example.
Example 11.1.4.
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1. Write x(t) = 5e−t/5 cos(t) + 5e−t/5 3 sin(t) in the form x(t) = A(t) sin(ωt + φ). Graph x(t)
using a graphing utility.
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2. Write x(t) = (t + 3) 2 cos(2t) + (t + 3) 2 sin(2t) in the form x(t) = A(t) sin(ωt + φ). Graph
x(t) using a graphing utility.
3. Find the period of x(t) = 5 sin(6t) − 5 sin (8t). Graph x(t) using a graphing utility.
Solution.
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1. We start rewriting x(t) = 5e−t/5 cos(t√+ 5e−t/5 3 sin(t) by factoring out 5e−t/5 from both
)
terms to get x(t) = 5e−t/5 cos(t) + 3 sin(t) . We convert what’s left in parentheses to
the required form using the formulas introduced in Exe...

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