About Trigonometry
Trigonometric Angles. This topic of trigonometry explains the difference between a geometric angle and a trigonometric angle. Right angles, straight angles, and revolutions are then defined. Graphing angles in the Cartesian coordinate system is described by defining the standard position of an angle and then explaining when an angle is positive and when an angle is negative. Finally, coterminal angles are detailed.
Measure of Angles. Angles can be measured in revolutions, degrees, and radians. This topic of trigonometry defines degrees and radians and shows how to convert between them. Angles measured in degrees can also be repesented in decimal notaton and degree-minute-second notation. Examples are given for each representation.
Unit Circle. The unit circle is a circle of radius 1. The unit circle is used in trigonometry as a mnemonic device for remembering special angles in degrees and radians and the trigonometric values of these angles. This topic of trigonometry describes how to construct unit circles by showing the steps that can be used in constructing one.
Arc Length. This topic of trigonometry introduces the arc length subtended by a central angle and the area of a wedge cut out by a central angle. Examples are given which emphasize that the central angle must be in radians. Finally, a relationship between arc length and the area of a sector is given.
Similiar Triangles and Right Triangles. The idea of similar triangles has been around for thousands of years and is present in Euclid's book The Elements. Similar triangles are important for working with triangles but the importance also lies in the fact that similar triangles allow us to define the trigonometric functions. In this topic we explain similar triangles and state the Pythagorean Theorem and its converse. Succinctly, the Pythagorean Theorem states: in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. The converse is also true: if the square of the length of the longest side is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.
Introducing the Trigonometric Functions. Similar triangles is the geometric idea that is used to define the trgionometric functions as ratios of sides of right triangles. Given an angle there are six possibilites for a function to be defined. The trigonometric functions sine, cosine, tangent, cotangent, secant, and cosecant of an acute angle are defined and detailed with examples and comments. Mnemonic devices are given for the definitions and for the functional values for the special angles of 30, 45, and 60 degrees. The reciprocal properties explain that three of the six trigonometric functions (sine, cosine, and tangent) are just reciprocals of three other trigonometric functions (cosecant, secant, and cotangent). Thus, if the values of some of the trigonometric functions are known the values of the others might be easily found. Finally, the Pythagorean Identities allows us to find the values of the six trigonometric functions given only one of the values.
Trigonometric Functions of a Single Angle. The trigonometric functions for any angle are defined and it is shown how to evaluate the trigonometric functions of an angle whose terminal side passes through a given point. The relationships between the trigonometric functions of an angle and the trigonometric functions of the negative of the angle are given. Reference angles are then defined and it is illustrated how to use the reference angle to evaluate the six trigonometric functions.
Graphs of Sine and Cosine. This topic of trigonometry presents the graphs of the sine and cosine functions. These graphs can be obtained by plotting points or using a knowledge of the values of the trigonometric functions, reference angles, and symmetry. The basic transformations of the sine and cosine are detailed including the amplitude (height), the period (portion that repeats), and the phase shift (horizontal shift).
Graphs of Secant and Cosecant. This topic of precalculusc presents the graphs of the secant and cosecant functions. These graphs can be obtained by plotting points or using a knowledge of the values of the trigonometric functions, reference angles, and symmetry. The basic transformations of the secant and cosecant are detailed including the amplitude (height), the period (portion that repeats), and the phase shift (horizontal shift).
Graphs of Tangent and Cotangent. This topic of trigonometry presents the graphs of the tangent and cotangent functions. These graphs can be obtained by plotting points or using a knowledge of the values of the trigonometric functions, reference angles, and symmetry. The basic transformations of the tangent and cotangent are detailed including the period (portion that repeats) and the phase shift (horizontal shift).
Additional Trigonometric Graphs. This topic of trigonometry presents the graphs of trigonometric functions with some type of transformation applied to it. For example, applying an absolute value function to a trigonometric function yields a similarly shape function by relecting the negative portion of the graph through the y-axis. Other types of transformations include, vertical transformations, sums and differences, and damping effects.
Trigonometric Identities. Algebraic formulas along with the trigonometric definitions and the Pythagorean Identities can produce quite a number of trigonometric identities. A beginning strategy for proving a trigonometric identity is to reduce the more complicated side of the equation to the simpler side. If no simplification is obvious, then perhaps express every trigonometric function on the more complicated side of the equation in terms of sines and cosines, and then reduce this expression to the simpler side.
Trigonometric Equations. A trigonometric equation that is true for only some values of the variable but not for others is called a conditional trigonometric equation. In this topic of trigonometry we give examples of solving different types of trigonometric equations. Some trigonometric equations can be solved by factoring and others by using trigonometric identities. One technique is to square both sides which used be used with caution. In this case, checking for extraneous solutions is not optionally.
Cite this as:About Trigonometry
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/about-trigonometry.html
