Trigonometry

Abstract

<div class="abstract" data-abstract-type="normal"> <span class='bold'>Introduction</span> This chapter begins the consideration of mathematical topics expected to be known for the AMC 12 exam but not for the AMC 10. There have not been many problems involving trigonometry on the more recent exams because the use of calculators makes many of the traditional problems trivial. However, the topic is important and the subject matter dealing with this subject is quite general. Students taking the AMC 10 examinations will not see problems involving trigonometry. <span class='bold'>Definitions and Results</span> The two very basic definitions in trigonometry are the sine and the cosine of a given number or given angle. There are two standard and equivalent ways to define these concepts; one uses right triangles, and the other uses the unit circle. Defining the sine and cosine for angles using right triangles is generally the first definition that is presented, but the unit circle approach is more appropriate when the sine and cosine are needed for functional representation. We will give the unit circle definition, since it is more general, and may not be as familiar. Place a unit circle in the <span class='italic'>xy</span>-plane. For each positive real number <span class='italic'>t</span>, define <span class='italic'>P(t)</span> as the point on this unit circle that is a distance <span class='italic'>t</span> along the circle, measured counterclockwise, from the point (1, 0). For each negative number <span class='italic'>t</span>, define <span class='italic'>P(t)</span> as the point on this unit circle that is a distance |<span class='italic'>t</span>| along the circle, measured clockwise, from the point (1, 0). Finally, define <span class='italic'>P(</span>0) = (1, 0). In this way we have, for each real number <span class='italic'>t</span>, a unique pair (<span class='italic'>x(t), y(t))</span> of <span class='italic'>xy</span>-coordinates on the unit circle to describe the point <span class='italic'>P(t)</span>. These coordinates provide the two basic trigonometric functions. <span class='bold'>Definition 1 The Sine and Cosine:</span> Suppose that the coordinates of a point <span class='italic'>P(t)</span> on the unit circle are (<span class='italic'>x(t), y(t))</span>. Then the <span class='bold'>sine of <span class='italic'>t</span></span>, written sin <span class='italic'>t</span>, and the <span class='bold'>cosine of <span class='italic'>t</span></span>, written cos <span class='italic'>t</span>, are defined by sin <span class='italic'>t</span> = <span class='italic'>y(t)</span> and cos <span class='italic'>t</span> = <span class='italic'>x(t).</span> These definitions are also used for the sine and cosine of an angle with radian measure <span class='italic'>t</span>. So the trigonometric functions serve two purposes, directly as functions with domain the set of real numbers and indirectly as functions with domain the set of all possible angles, where the angles are given in radian measure. </div>