1. Trigonometric Functions: The Unit Circle MATH 109 - Precalculus S. Rook

2. Overview Section 4.2 in the textbook: – Circular trigonometric functions – Properties of sine & cosine – Even & odd functions 2

3. Circular Trigonometric Functions

4. Unit Circle Unit Circle: a special circle with a radius of 1, center of (0, 0), and equation of x 2 + y 2 = 1 4

5. Real Number Line & Unit Circle 5 Starting at (1, 0) consider “wrapping” the positive real number line around the unit circle – Each number on the real number line corresponds to ONE point on the unit circle – Each point on the unit circle corresponds to MANY points on the real number line Recall that the circumference of a circle is 2πr Thus, each revolution around the unit circle constitutes 2π ≈ 6.28 units

6. Circular Functions Now consider a point (x, y) on the circumference of the unit circle where t is the length of the arc from (1, 0) to (x, y) Then – The central angle is equivalent to the length of the arc it cuts on the Unit Circle 6

7. Circular Functions (Continued) All points encountered on the unit circle can then be written as a function of t where t is the distance traveled from (1, 0) t can be positive (counterclockwise) or negative (clockwise) There are six trigonometric functions – defined with respect to the Unit Circle they are: 7

8. Circular Functions (Continued) We call these the circular functions – The radian measure of θ is the same as the arc length from (1, 0) to a point P on the terminal side of θ on the circumference of the unit circle Any point (x, y) on the unit circle can be written as (cos t, sin t) It is to your advantage to memorize the values in Quadrant I and the quadrantal angles (next slide) 8

9. Common Angles and Function Values on the Unit Circle DegreesRadianscos θsin θ 0°010 30° 45° 60° 90°01 180°0 270°0 360°10 9

10. Circular Functions (Example) Ex 1: Use the Unit Circle to find the six trigonometric functions of: a) b) 10

11. Circular Functions (Example) Ex 2: Use the Unit circle to find all values of t, 0 < t < 2π where a) b) c) 11

12. Circular Functions (Example) Ex 3: If t is the positive distance from (1, 0) to point P along the circumference of the unit circle, sketch t on the circumference of the unit circle and then find the value of: a)P = (0.8560, -0.5169); find i) cos t, ii) csc t, and iii) cot t b)P = (0.0432, 0.9782); find i) sin t, ii) sec t, and iii) tan t 12

13. Properties of Sine & Cosine

14. Domain & Range of Sine & Cosine Recall that domain is the allowable set of input values for a function: – Given (x, y) = (cos t, sin t) on the unit circle, there are no places on the unit circle where cos t or sin t are undefined Domain of f(t) = sin t and f(t) = cos t is (-oo, +oo) Recall that range is the acceptable output values for a function: – All points (x, y) = (cos t, sin t) on the unit circle must satisfy: -1 ≤ (x, y) ≤ 1 Range of f(t) = sin t and f(t) = cos t is [-1, 1] 14

15. Period of Sine & Cosine We have already seen that ONE revolution around the unit circle occurs when t assumes values in the interval [0, 2π) Given a function f(t), the period is the smallest value c, c > 0 such that f(t + c) = f(t) for all t in the domain of f – i.e. when the function values start to repeat Given the point t = (x, y) on the unit circle, what value added to t results in the same point (x, y)? – Thus: sin(t + 2πn) = sin(t) and cos(t + 2πn) = cos(t), n is an integer – What is the period of f(t) = sin t and f(t) = cos t 2π2π 15

16. Period of Sine & Cosine (Example) Ex 4: Evaluate the trigonometric function using its period [rewrite in the form cos(t + 2πn) or sin(t + 2πn)] and the unit circle: a) b) 16

17. Even and Odd Functions

18. Recall the definition of even and odd functions: – If f(-t) = f(t), f(t) is an even function – If f(-t) = -f(t), f(t) is an odd function Examine the Unit Circle at the right: cos(-θ) = cos θ meaning? sin(-θ) = -sin θ meaning? EvenOdd 18

19. Even & Odd Functions (Example) Ex 5: Use to evaluate: a)cos t b)sec(-t) 19

20. Summary After studying these slides, you should be able to: – Define the six trigonometric functions in terms of the unit circle – Given a value of t or a point (x, y) evaluate the six trigonometric functions – State the domain, range, and period of the sine and cosine – Understand even & odd functions Additional Practice – See the list of suggested problems for 4.2 Next lesson – Right Triangle Trigonometry (Section 4.3) 20