Trigonometric Functions of Real Numbers

All trigonometric definitions and identities previously used with the angle hold true for radian measure or real number measure t. The letter t may be used to represent the radian measure of the angle generated by an initial ray and terminal ray.

If the definitions of the trigonometric functions are based on the unit circle, then r = 1 and . From this relationship the generated angle is the same as the length of the arc subtended by the angle. Thus if and , then the arc length, s is equal to t.

Pythagorean Identities

Ex 2. Show that

Table1

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Opposite Angles Identities

cos(-t) = cos(t)

sin(-t) = -sin(t)

tan(-t)= -tan(t)

Note:

· The cosine function has y-axis symmetry and is an even function. All even functions have y-axis symmetry and satisfy the relationship f(-x) = f(x).

• The sine and tangent functions have origin symmetry and are odd functions. All odd functions have origin symmetry and satisfy the relationship f(-x) = -f(x).

Periodicity of sine and cosine

More generally, for any real number t and any integer k, the following identities hold: