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Consider the sine function as $\"\"$ and the value of $\"\"$ is $\"\"$ .

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Linearization $\"\"$ of the function is at $\"\"$ is $\"\"$ .

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$\"\"$ .

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Differentiate on each side with respect to $\"\"$ .

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$\"\"$

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Substitute $\"\"$ in the function.

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$\"\"$

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$\"\"$ .

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$\"\"$ .

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Linearization $\"\"$ of the function is at $\"\"$ is $\"\"$ .

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Substitute corresponding values in linear approximation equation.

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$\"\"$

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Therefore linear approximation for the sine function is $\"\"$ .

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(b)

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The values of $\"\"$ and $\"\"$ differ by $\"\"$ % represents the relative error is $\"\"$ %.

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Relative error $\"\"$ .

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Graph:

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Graph the % error.

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$\"\"$

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Observe the graph.

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The two functions differ by $\"\"$ % at $\"\"$ .

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$\"\"$ .

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$\"\"$ .

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The two functions differ by $\"\"$ % at $\"\"$ .