Estimate sin(0.05) with a linear approximation

Step by step

The question

Question: estimate sin 0.05 with a linear approximation.

The function is f(x) = sin x.

Near 0.05 there is a friendly number: a = 0.

sin 0 is 0.

1 The point

Put 0 into f.

We get 0.

The point is (0, 0).

2 The slope there

The slope is the derivative: f′(x) = cos x.

At x = 0, the slope is 1.

3 The tangent line

The tangent line: start at the height f of a, and go up by the slope times the distance from a.

So L(x) = x.

Let's see it.

The curve.

And the tangent line at x = 0.

Near 0, they are almost the same.

4 Estimate

Put 0.05 into L.

Careful!

Use the distance from 0: 0.05 minus 0.

Not 0.05 itself.

0.05 minus 0 is 0.05.

So sin 0.05 ≈ 0.05.

5 Check

Let's check with a calculator in radians: sin 0.05 is 0.04998 and a bit.

Our estimate 0.05 is very close.

Right!

Let's look back.

First: the point.

Then: the slope there.

Then: the tangent line.

Then: estimate.

Last: check.