Video: Linear Approximation of the Sine Function

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

02:44

Video Transcript

By finding the linear approximation of the function 𝑓 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 𝑓 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 we’re trying to estimate. Since we’re trying to estimate the sin of 3.14, we should pick a value near 3.14. So, we’ll choose our linear approximation about the point π‘₯ is equal to πœ‹. We recall, if a function 𝑓 is differentiable at the point π‘₯ is equal to π‘Ž, then the tangent line 𝐿 of 𝑓 at π‘Ž is equal to 𝑓 evaluated at π‘Ž plus 𝑓 prime evaluated at π‘Ž multiplied by π‘₯ minus π‘Ž.

We use the tangent line as our linear approximation. If we set 𝑓 to be equal to the sin of π‘₯ and we set our point π‘Ž to be equal to πœ‹, then our linear approximation of the sine function about πœ‹ is given by the sin of πœ‹ plus the derivative of sin evaluated at πœ‹ multiplied by π‘₯ minus πœ‹. 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œ‹ is cos of πœ‹. Putting this together gives us that the tangent line of the sine function at πœ‹ is equal to the sin of πœ‹ plus the cos of πœ‹ multiplied by π‘₯ minus πœ‹.

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

And we can see that our line and our curve are very close around π‘₯ is equal to πœ‹. 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œ‹ minus 3.14. Therefore, we’ve shown by using a linear approximation of the sine function about πœ‹, we can estimate the value of the sin of 3.14 to be πœ‹ minus 3.14.

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