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(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.

θ0.10.20.30.40.5
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:

θ0.10.20.30.40.5
sin θ0.09980.1987 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}.


Learn more about this topic:

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Trigonometry: Sine and Cosine

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Chapter 4 / Lesson 6
40K

Sine and cosine are basic trigonometric functions that are used to solve for the angles and sides of triangles. Review trigonometry concepts and learn about the mnemonic used for sine, cosine, and tangent functions.


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