5. If position of object x = 3 sinΘ – 7 cosΘ then motion of object is bounded between position.​

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Answer:

  ±√58 ≈ ±7.616

Step-by-step explanation:

The linear combination of sine and cosine functions will have an amplitude that is the root of the sum of the squares of the individual amplitudes.

  |x| = √(3² +7²) = √58

The motion is bounded between positions ±√58.

_____

Here's a way to get to the relation used above.

The sine of the sum of angles is given by ...

  sin(θ+c) = sin(θ)cos(c) +cos(θ)sin(c)

If this is multiplied by some amplitude A, then we have ...

  A·sin(θ+c) = A·sin(θ)cos(c) +A·cos(θ)sin(c)

Comparing this to the given expression, we find ...

  A·cos(c) = 3   and   A·sin(c) = -7

We know that sin²+cos² = 1, so the sum of the squares of these values is ...

  (A·cos(c))² +(A·sin(c))² = A²(cos(c)² +sin(c)²) = A²(1) = A²

That is, A² = (3)² +(-7)² = 9+49 = 58. This tells us the position function can be written as ...

  x = A·sin(θ +c) . . . . for some angle c

  x = (√58)sin(θ +c)

This has the bounds ±√58.

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