# (a) Complete the table. |θ|0.1|0.2|0.3|0.4|0.5 |sin θ| | | | | (b) Is θ or sin...

## Question:

(a) Complete the table.

 θ 0.1 0.2 0.3 0.4 0.5 sin θ

(b) Is θ or sin θ greater for θ in the interval (0, 0.5]?

(c) As θ approaches 0, how do θ and sin θ compare? Explain.

## Small-Angle Approximation:

The small-angle approximation is an approximation technique used to simplify expressions that have sine functions in them. Basically, if the angle is really small, then we can make the following assumption:

{eq}\displaystyle \sin\theta\ \approx\ \theta {/eq}

This approximation, of course, will not work for large values of {eq}\displaystyle \theta {/eq}.

## Answer and Explanation: 1

(a) Let us first assume that the angles here are in radians. The values are:

 θ 0.1 0.2 0.3 0.4 0.5 sin θ 0.0998 0.1987 0.2955 0.3894 0.4794

(b) As {eq}\displaystyle \theta\to 0.5, {/eq} then value of {eq}\displaystyle \theta > \sin\theta {/eq}

(c) As {eq}\displaystyle \theta\to 0 {/eq}, the values of {eq}\displaystyle \sin\theta {/eq} and {eq}\displaystyle \theta {/eq} become more and more similar. This is the basis behind the small-angle approximation {eq}\displaystyle (\sin\theta\approx\theta) {/eq}.