#### To determine

**To verify:** The linear approximation to the sine function at x=0

#### Answer

The linear approximation of sine function at x=0 is, L(x)=x.

#### Explanation

**Given:**

The value f(x)=sinx.

**Result used:**

The linear approximation of the function at x=a is,f(x)≈f(a)+f′(a)(x−a).

**Calculation:**

The linearization of the function f(x) at x=0 is computed as follows,

Substitute the value a=0 in f(x)≈f(a)+f′(a)(x−a),

f(x)≈f(0)+f′(0)(x)

The derivative of the function f(x)=sinx is, f′(x)=cosx.

Substitute x=0 in f(x),

f(0)=sin0=0

Substitute x=0 in f′(x)=cosx,

f′(0)=cos0f′(0)=1

Substitute f′(0)=1 and f(0)=0 in f(x)≈f(0)+f′(0)(x),

f(x)≈0+1(x)f(x)≈x

Thus, the linear approximation of sine function at x=0 is, L(x)=x.

#### To determine

The values of *x* for which sinx and x differ by less than 2% and verify that the Hecht’s statement by converting from radians to degrees.

#### Answer

The function sinx and x differ by less than 2% when x<0.347.

#### Explanation

**Calculation:**

The required value of *x* such that the sinx and x differ by less than 2%.

That is, y=x−sinxx<0.02

Using online graphing calculator to draw the function y=x−sinxx is shown below,

From Figure 1, it is clearly observed that the value y=x−sinxx is less than 0.02002 whenever the value of *x* is less than 0.347.

Since 1 radian=180°π .

0.347 radians=180°π×0.347=57.2958×0.347=19.88 degrees≈20 degrees

Therefore, the values of x≈20°.