By finding the linear approximation of the function 𝑓 of 𝑥 is equal to sin 𝑥 at a suitable value of 𝑥, 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 𝑥 is equal to sin 𝑥 at a suitable value of 𝑥. 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 𝑥 is equal to 𝜋. We recall, if a function 𝑓 is differentiable at the point 𝑥 is equal to 𝑎, then the tangent line 𝐿 of 𝑓 at 𝑎 is equal to 𝑓 evaluated at 𝑎 plus 𝑓 prime evaluated at 𝑎 multiplied by 𝑥 minus 𝑎.
We use the tangent line as our linear approximation. If we set 𝑓 to be equal to the sin of 𝑥 and we set our point 𝑎 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 𝑥 minus 𝜋. We recall the derivative with respect to 𝑥 of the sin of 𝑥 is equal to the cos of 𝑥. This gives us the derivative of the sin of 𝑥 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 𝑥 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 𝑥 minus 𝜋. We can simplify this again by distributing the negative one over our parentheses which gives us negative 𝑥 plus 𝜋, which we can rearrange to give us 𝜋 minus 𝑥. So, what we found is the tangent line to the sine graph when 𝑥 is equal to 𝜋. What this means is, if we were to sketch a graph of 𝑦 is equal to the sin of 𝑥, then we can sketch our graph of 𝐿 of 𝑥, which gives us a tangent line when 𝑥 is equal to 𝜋.
And we can see that our line and our curve are very close around 𝑥 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 𝑥 is equal to 3.14. And to calculate this, we just substitute 𝑥 is equal to 3.14 into our equation for the line. This gives us 𝜋 minus 3.14. Therefore, we’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.