4.2 Trigonometric Functions: The Unit Circle

The two historical perspectives of trigonometry incorporate different

The Unit circle: xy221

2 2   3 1   1 3  Example: Verify the points ,  ,  ,  ,  ,  ,(1,0) are on the unit circle. 2 2   2 2   2 2 

Imaging that the real number line is wrapped around this circle, with positive numbers corresponding to a counterclockwise wrapping and negative numbers corresponding to a clockwise wrapping, as shown in the following.

As the real number line is wrapped around the unit circle, each real number t corresponds to a point  (,)xy on the circle. For example, the real number corresponding to (0,1) . 2

Remark: In general, each real number t also corresponds to a central angle  (in standard position) whose radian measure is t . With this interpretation of t , the arc length formula sr  (with r 1 ) indicates that the real number t is the (directional) length of the arc intercepted by the angle  , given in radians. In the following graph, the unit circle has been divided into eight equal arcs, corresponding to t -values  3  5  3  7  of 0, , , , , , , ,2 4 2 4 4 2 4

Similarly, in the following graph, the unit circle has been divided into 12 equal arcs, corresponding to t    2  5  7  4  3  5  11  values of 0,,,, , ,, , , , , ,2 6 3 2 3 6 6 3 2 3 6

The Trigonometric Functions.

From the preceding discussion, it follows that the coordinates x and y are two functions of the real variable t . You can use these coordinates to define the six trigonometric functions of t .

sine (sin) cosecant (csc) cosine(cos) secant(sec) tangent(tan) cotangent(cot)

Definitions of Trigonometric Functions.

Let t be a real number and let (,)xy be the point on the unit circle corresponding to t .

y sint y cos t  x tan t  , x  0 x

11x csct , y  0 sec t  , x  0 cot= , y  0 y x y Example: Evaluate the six trigonometric functions at each real number:    3   t, t  , t  , t  , t  , t  , t  2 , t   6 4 3 2 4 6

Domain and period of Sine and Cosine.

The domain of the sine and cosine functions is the set of all real numbers. To determine the range of these two functions, we have

1  sintt  1,  1  cos  1 .

The values of sin(tn 2 ) and cos(tn 2 ) correspond to those of sintt and cos . Similar results can be obtained for repeated revolutions (positive or negative) on the unit circle. The leads to the general result sin(t 2 n ) sin t cos(t 2 n ) cos t for any integer n and real number t . Functions that behave in such a repetitive manner are called periodic.

The definition f is periodic if there exists a positive real number c such that

f()() x c f x for all t in the domain of f . The smallest number c for which f is periodic is called the period of f .

Even and Off Trigonometric Functions. cos(tt ) cos sin(tt )   sin tan(tt )   tan

Example: Using period and even and odd properties of trigonometric functions to evaluate

15   2  sin  ,cos   33   