Unit Circle Trigonometry

1. Pre-Calculus Day 39 Unit CircleTrigonometry

2. What You Should Learn Identify a unit circle and describe its relationship to real numbers. Evaluate trigonometric functions using the unit circle.

3. Plan for the day… • Review of homework • Quiz • Quick Review • Let’s build the Unit Circle… • Homework

4. Trigonometric Functions θ adj opp sin  = cos  = tan  = csc  = sec  = cot  = hyp adj hyp hyp adj opp adj opp hyp The trigonometric functions are opp adj sine, cosine, tangent, cotangent, secant, and cosecant. Note: sine and cosecant are reciprocals, cosine and secant are reciprocals and tangent and cotangent are reciprocals

5. Fundamental Trigonometric Identities Fundamental Trigonometric Identities for Cofunction Identities sin  = cos(90  ) cos  = sin(90  ) sin  = cos (π/2  ) cos  = sin(π/2  ) tan  = cot(90  ) cot  = tan(90  ) tan  = cot(π/2  ) cot  = tan(π/2  ) sec  = csc(90  ) csc  = sec(90  ) sec  = csc(π/2  ) csc  = sec(π/2  ) Reciprocal Identities sin  = 1/csc cos = 1/sec tan = 1/cotcot = 1/tan sec = 1/cos csc = 1/sin Quotient Identities tan  = sin  /cos  cot  = cos  /sin  Pythagorean Identities sin2  + cos2  = 1 tan2 + 1 = sec2 cot2  + 1 = csc2 

6. Name the Equivalent Function…

7. sin θ 1/csc

8. tan θ sin θ/cos

9. sec θ csc (90o-)

10. cot θ cos θ/sin

11. tan θ 1/cot

12. cos θ sin (90o -)

13. sec θ 1/cos

14. A circle defined by x2 + y2 = 1 Imagine the real number line wrapped around the circle. Each real number t corresponds to a point (x, y). Since the radius is 1, the number t would correspond with the central angle (s = rθ). The Unit Circle (0, 1) (x, y) t (-1, 0) (1, 0) θ= t (0, -1)

15. 45 1 45 Geometry of the 45-45-90 triangle Consider an isosceles right triangle with a hypotenuse the length of 1. What would be the length of the sides?

16. Example: Trig Functions for  30 1 60○ 30 Geometry of the 30-60-90 triangle Consider a 30-60-90 triangle with a hypotenuse the length of 1. What would be the length of the sides?

17. The Unit Circle • The values on the unit circle are obtained by overlaying the special right triangles in each quadrant. • Because the hypotenuse of each triangle has a length of 1, and starts at the origin, then the coordinates of the endpoint of the hypotenuse correlate with the lengths of the adjacent and opposite sides. • Likewise due to the hypotenuse length of 1, the cos and sin of each angle reduces to adj and hyp, respectively. • Therefore, the coordinates of the endpoint of the hypotenuse are (cos, sin ).

19. Definitions of Trig Functions As mentioned, a real number t will correspond to a point (x, y) on the unit circle, and the angle which forms that arc will have a measure of t radians.

20. Try this Evaluate the six trigonometric functions at each real number: a) t = π/6 b) t = 5π/4 c) t = 0 d) t = π e) t = - π/6 f) t = 13π/6

21. Domain of Sine and Cosine • The domain of the sine and cosine functions is the set of all real numbers. • To find the range, consider the unit circle. Remember that sin t = y and cos t = x. • The range for sin and cos is [-1, 1].

22. Periodic Functions • A function is periodic if there exists a positive real number c such that:for all t in the domain of f. • The smallest number c for which f is periodic is called the period of f. • Because the outputs of sin and cos repeat every full rotation, the period for these functions is 2π (or 360).

23. Even and Odd Trig Functions • Cosine and secant functions are even cos (-t) = cos t sec (-t) = sec t • Sine, cosecant, tangent and cotangent are odd sin (-t) = - sin t csc (-t) = - csc t tan (-t) = - tan t cot (-t) = - cot t

24. Homework 21 Section 4.2 pp. 278-279 # 1-39 odd, 45, 49 Memorize your unit circle!!! Quiz after break!!