# Question Video: Linear Approximation of the Sine Function Mathematics • Higher Education

By finding the linear approximation of the function π(π₯) = sin π₯ at a suitable value of π₯, estimate the value of sin (3.14).

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### Video Transcript

By finding the linear approximation of the function π of π₯ is equal to sin π₯ at a suitable value of π₯, estimate the value of the sin of 3.14.

The question wants us to estimate the value of the sin of 3.14 by finding a linear approximation of the function π of π₯ is equal to sin π₯ at a suitable value of π₯. We recall that linear approximations are best when we pick a value near what weβre trying to estimate. Since weβre trying to estimate the sin of 3.14, we should pick a value near 3.14. So, weβll choose our linear approximation about the point π₯ is equal to π. We recall, if a function π is differentiable at the point π₯ is equal to π, then the tangent line πΏ of π at π is equal to π evaluated at π plus π prime evaluated at π multiplied by π₯ minus π.

We use the tangent line as our linear approximation. If we set π to be equal to the sin of π₯ and we set our point π to be equal to π, then our linear approximation of the sine function about π is given by the sin of π plus the derivative of sin evaluated at π multiplied by π₯ minus π. We recall the derivative with respect to π₯ of the sin of π₯ is equal to the cos of π₯. This gives us the derivative of the sin of π₯ evaluated at π is cos of π. Putting this together gives us that the tangent line of the sine function at π is equal to the sin of π plus the cos of π multiplied by π₯ minus π.

We could simplify this further. We know that the sin of π is just equal to zero and the cos of π is equal to negative one. This gives us zero plus negative one multiplied by π₯ minus π. We can simplify this again by distributing the negative one over our parentheses which gives us negative π₯ plus π, which we can rearrange to give us π minus π₯. So, what we found is the tangent line to the sine graph when π₯ is equal to π. What this means is, if we were to sketch a graph of π¦ is equal to the sin of π₯, then we can sketch our graph of πΏ of π₯, which gives us a tangent line when π₯ is equal to π.

And we can see that our line and our curve are very close around π₯ is equal to π. So, we can use our line to approximate the sin of 3.14. So, using linear approximation gives us the sin of 3.14 is approximately equal to our line evaluated at π₯ is equal to 3.14. And to calculate this, we just substitute π₯ is equal to 3.14 into our equation for the line. This gives us π minus 3.14. Therefore, weβve shown by using a linear approximation of the sine function about π, we can estimate the value of the sin of 3.14 to be π minus 3.14.