Trigonometric Functions: The Unit Circle 4.2

1. Trigonometric Functions: The Unit Circle4.2

2. Unit Circle • The unit circle is a circle of radius 1 with its center at the origin.

3. If t is a real number and P = (x, y) is a point on the unit circle that corresponds to t, then Definitions of the Trigonometric Functions in Terms of a Unit Circle

4. Points on the Unit Circle

5. Example Use the Figure to find the values of the trigonometric functions at t=/2. Solution: The point P on the unit circle that Corresponds to t= /2 has coordinates (0,1). We use x=0 and y=1 to find the Values of the trigonometric functions

6. The Domain and Range of the Sine and Cosine Functions • The domain of the sine function and the cosine function is the set of all real numbers • The range of these functions is the set of all real numbers from -1 to 1, inclusive.

7. Evaluating Trigonometric Functions • Evaluate the six trig functions at each real number. (a) t=л/6 (b) t=5л/4 (c) t=0 (d) t=л

8. Evaluate the 6 Trig Functions at t=-л/3

9. The cosine and secant functions are even. cos(-t) = cos t sec(-t) = sec t The sine, cosecant, tangent, and cotangent functions are odd. sin(-t) = -sin t csc(-t) = -csc t tan(-t) = -tan t cot(-t) = -cot t Even and Odd Trigonometric Functions

10. Example • If sin t = 2/5 and cos t = 21/5, find the remaining four trig functions

11. A function f is periodic if there exists a positive number p such that f(t + p) = f(t) For all t in the domain of f. The smallest number p for which f is periodic is called the period of f. Definition of a Periodic Function

12. sin(t + 2) = sin t and cos(t + 2) = cos t The sine and cosine functions are periodic functions and have period 2. Periodic Properties of the Sine and Cosine Functions sin  = sin 3

13. tan(t + ) = tan t and cot(t + ) = cot t The tangent and cotangent functions are periodic functions and have period . Periodic Properties of the Tangent and Cotangent Functions tan  = tan 2