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11: Applications of Trigonometry

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    • 11.1: Applications of Sinusoids
      In the same way exponential functions can be used to model a wide variety of phenomena in nature, the cosine and sine functions can be used to model their fair share of natural behaviors
    • 11.2: The Law of Sines
      Trigonometry literally means 'measuring triangles', we are more than prepared to do just that. The main goal of this section and the next is to develop theorems which allow us to 'solve' triangles -- that is, find the length of each side of a triangle and the measure of each of its angles.
    • 11.3: The Law of Cosines
      The Law of Sines to enable us to solve triangles in the 'Angle-Angle-Side' (AAS), the 'Angle-Side-Angle' (ASA) and the ambiguous 'Angle-Side-Side' (ASS) cases. In this section, we develop the Law of Cosines which handles solving triangles in the 'Side-Angle-Side' (SAS) and 'Side-Side-Side' (SSS) cases.
    • 11.4: Polar Coordinates
      Cartesian coordinates of a point are often called 'rectangular' coordinates. In this section, we introduce a new system for assigning coordinates to points in the plane -- polar coordinates. We start with an origin point, called the pole, and a ray called the polar axis. We then locate a point PP using two coordinates, (r,θ), where r represents a directed distance from the pole.
    • 11.5: Graphs of Polar Equations
      In this section, we discuss how to graph equations in polar coordinates on the rectangular coordinate plane.
    • 11.6: Hooked on Conics Again
      In this section, we revisit our friends the Conic Sections which we began studying in Chapter 7. Our first task is to formalize the notion of rotating axes.  Armed with polar coordinates, we can generalize the process of rotating axes as shown below.
    • 11.7: Polar Form of Complex Numbers
      In this section, we return to our study of complex numbers. We associate each complex number z=a+biz=a+bi with the point (a,b)(a,b) on the coordinate plane. In this case, the xx -axis is relabeled as the real axis, which corresponds to the real number line as usual, and the yy -axis is relabeled as the imaginary axis, which is demarcated in increments of the imaginary unit ii . The plane determined by these two axes is called the complex plane.
    • 11.8: Vectors
      To answer questions that involve both a quantitative answer, or magnitude, along with a direction, we use the mathematical objects called vectors. The word 'vector' comes from the Latin vehere meaning 'to convey' or 'to carry.' A vector is represented geometrically as a directed line segment where the magnitude of the vector is taken to be the length of the line segment and the direction is made clear with the use of an arrow at one endpoint of the segment.
    • 11.9: The Dot Product and Projection
      Previously, we learned how add and subtract vectors and how to multiply vectors by scalars. In this section, we define a product of vectors.
    • 11.10: Parametric Equations
      There are scores of interesting curves which, when plotted in the xy-plane, neither represent y as a function of x nor x as a function of y. In this section, we present a new concept which allows us to use functions to study these kinds of curves.

    This page titled 11: Applications of Trigonometry is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Carl Stitz & Jeff Zeager via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.