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Trigonometry

Trigonometric Functions and Identities

he trigonometric functions are among the most fundamental in mathematics. They were initially developed to aid in the measurement of triangles and their angles, and they are useful in such practical fields as surveying and navigation. However, their importance to pure and applied mathematics extends far beyond these uses, because they can be used to describe any natural phenomenon that is periodic, and in higher mathematics they are fundamental tools for understanding many abstract spaces.

The primary use of trigonometric functions is in the measurement of angles. Although the ancient Babylonian degree unit of angle measure is still in wide use, in mathematics we prefer to use the radian measure. Given a circle centered at the origin in the Cartesian plane, imagine taking a radius and laying it along the outside circle, beginning at the x axis and going counterclockwise.

This marks out an angle of one radian. Because the circumference of a circle is twice the radius times p, a full circle corresponds to an angle of 2p radians. Thus we get the following correspondences between degree measure and radian measure:

By remembering the correspondence between radians and degrees indicated by the formula above, one may always convert radians to degrees and vice versa with ease by plugging the known quantity into the equation and solving for the unknown quantity. With practice, using radian measure becomes as natural as using degrees, and the use of radians greatly simplifies our work with the trigonometric functions.

TRIGONOMETRIC FUNCTIONS

The trigonometric functions are most easily understood in the context of a circle in the Cartesian plane, in which angles are always measured from the x axis: positive angles are measured in an anti-clockwise direction, and negative angles are measured in a clockwise direction. In this context, the trig functions are defined as follows:

Note that these definitions induce the following relationships:

TRIGONOMETRIC IDENTITIES

The so-called trigonometric identities are actually quite easy to derive algebraically from the definitions given above, and every student of mathematics should derive them all at least once. Thereafter, it will no longer be necessary to memorize them, for you will be able to derive them when they are needed.

Pythagorean Identities

cofunction identities

reduction formulas

sum and difference formulas

double-angle formulas

power reducing formulas

sum-to-product formulas

product-to-sum formulas

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