Further Applications of Trigonometry
The world’s largest tree by volume, named General Sherman, stands 274.9 feet tall and resides in Northern California. Just how do scientists know its true height? A common way to measure the height involves determining the angle of elevation, which is formed by the tree and the ground at a point some distance away from the base of the tree. This method is much more practical than climbing the tree and dropping a very long tape measure. In this chapter, we will explore applications of trigonometry that will enable us to solve many different kinds of problems, including finding the height of a tree. We extend topics we introduced in Trigonometric Functions and investigate applications more deeply and meaningfully.
Sections
- Non-right Triangles - Law of Sines — use the Law of Sines to solve oblique triangles in the ASA, AAS, and ambiguous SSA cases, find the area of an oblique triangle from two sides and the included angle, and solve applied elevation problems.
- Non-right Triangles - Law of Cosines — use the Law of Cosines to solve SAS and SSS triangles, apply it to navigation and surveying problems, and compute areas with Heron’s formula.
- Polar Coordinates — plot points on a polar grid, convert coordinates and equations between polar and rectangular form, and identify the curves polar equations describe.
- Polar Coordinates: Graphs — test polar equations for symmetry, find zeros and maxima, and graph circles, cardioids, limaçons, lemniscates, rose curves, and Archimedes’ spirals.
- Polar Form of Complex Numbers — plot complex numbers, find absolute value, convert between rectangular and polar form, and use De Moivre’s Theorem for products, quotients, powers, and roots.
- Parametric Equations — build tables of plane-curve values, eliminate the parameter to recover rectangular equations, and construct parametrizations of curves.
- Parametric Equations: Graphs — graph plane curves from parametric equations with their orientation, relate them to rectangular graphs, and model projectile motion parametrically.
- Vectors — represent vectors geometrically and in component or unit-vector form, add, subtract, and scale them, find magnitude and direction, compute dot products, and solve applied velocity problems.
The sections above are adapted from Precalculus 2e, Chapter 8 by Jay Abramson, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org.