1. Unit 8 Trigonometric Functions Radian and degree measure Unit Circle Right Triangles Trigonometric functions Graphs of sine and cosine Graphs of other Trigonometric functions Inverse Trigonometric functions Applications and Models

2. 13.1 Right Triangle Trigonometry Evaluate trigonometric functions of acute angles Use the fundamental trigonometric identities Use trigonometric functions to model applied problems

3. Opp. hypotenuse Remember the trigonometric functions used to solve right triangles? Adjacent

4. In addition to the trigonometric functions we have already learned, we have three more. Notice they are merely the reciprocals of the three we already have.

5. A B C 4 3 5 Find the six trig functions of angle A:

6. It is also helpful to remember the special right triangle relationships Now with all of this information, we are able to find trigonometric values of many angles on the unit circle.

7. Inverse Sine Function sin -1 x or arcsin x Inverse Cosine Function cos -1 x or arcos x tan-1 x or arctan x Inverse Tangent Function Inverse trigonometric functions help us to find the angle when we are given the triangle ratios What this question is asking is, “what is the angle that has a cosine of ?”

8. x 5 25˚ Solve for x. Sketch a triangle that has an acute angle , and then find . 5 8

10. If you look up from the middle of the football field to the top of the gym, you see the angle of elevation is 41 degrees. If you are 250 feet from the base of the gym, how tall is the gym? 41 250 feet x 1) Draw picture first! 2) Set up trig ratio 3) Solve

11. 5150 x y An airplane is flying at an elevation of 5150 ft, directly above a straight highway. Two motorists are driving cars on the highway on opposite sides of the plane, and the angle of depression to one car is 35  and to the other is 52 . How far apart are the cars? 11378.58 feet 52 ˚35 ˚52 ˚ 35 ˚

12. 13.2 Radian and Degree Measure Describe angles Use radian measure Use degree measure Use angles to model applied problems

13. Positive angle Negative angle Terminal side of angle Initial side of angle This is known as standard position

14. Alpha and Beta have the same initial and terminal sides. Angles like this are called coterminal. To find positive or negative coterminal angles: Add or Subtract 360  or 2 . Coterminal Angles: 2 angles who have the same terminal sides.

15. A radian is the measure of the angle that intercepts an arc whose length is equal to the radius of the circle.

16. A quick conversion guide: Reminder- supplementary angles are two angles that add up to 180 degrees or radians. Complimentary angles are two angles that add up to 90 degrees or radians

17. Find the radian measure of the angle: Find the degree measure of the angle: a)b) Find 1 positive & 1 negative angle that are coterminal with the angle: a) b) Are the angles coterminal? no

18. Arc Length In radians s r Find s if the angle is 2 radians and the radius is 5cm s = (5)(2) = 10 cm Find the length of an arc that subtends a central angle of 45  in a circle of radius 10 m.

19. A sector of a circle of radius 24 mi has an area of 288 mi 2. Find the central angle of the sector. Area of a Circular Sector: A = sector area r = radius  = angle measure (rad) r

21. 13.6 Trigonometric functions: The Unit Circle Identify a unit circle and its relationship to real numbers Evaluate trigonometric functions using the unit circle Use the domain and period to evaluate sine and cosine functions Use a calculator to evaluate trigonometric functions

22. OP = 1 OR = x RP = y o p (x, y) R (1,0) (0,-1) (-1,0) (0,1)

23. O (1,0) (0,-1) (-1,0) (0,1) This means the point P at (x, y) can be represented by the ordered pair (cos x, sin x). It is easy for us to figure out the trigonometric functions if the terminal side of an angle falls on an axis.

24. It is also helpful to remember the special right triangle relationships Now with all of this information, we are able to find trigonometric values of many angles on the unit circle.

26. What is the sine, cosine, tangent of 150 degrees? A reference angle is the acute angle formed between the terminal side and the x-axis

27. Cosine also has a period of Tangent has a period of The sine function repeats itself every radians (or 360 ). The smallest number in which the function repeats itself is called the period. So, the sine function has a period of

28. 13.3 Trigonometric functions of any angle Evaluate trigonometric functions of any angle Use reference angles to evaluate trigonometric functions Evaluate trigonometric functions of real numbers

29. o p (x, y) OP = r Let us re-visit the circle of radius r. We can still find the length of x and y for any angle using trig. x y What is the length of y? What is the length of x?

30. This information allows us to find the sine and cosine for any angle.

31. o ALL I IV II III The positive trig. functions are listed. tangent cosine sine All-all Students-sine Take-tangent Calculus-cosine Use the reference angle formed between the terminal side and the x-axis

32. Find the reference angle for the given angle:

33. 1.Draw a picture 2.Find  (ref angle) 3.Trig (  ) = ±Trig (  ) 4.Decide if it’s POS or NEG Evaluating Trig Functions: Find the value of the other 5 trig functions of  given the following: We know it is in 4 th quadrant 5 1

34. 1 (1, 0) (0, 1) (0, -1) (-1, 0)

36. 13.7/13.8 Graphs of trig functions Sketch the graphs of basic sine and cosine functions Use amplitude and period to help sketch the graphs of sine and cosine functions Sketch the translations of the graphs of sine and cosine functions Use the sine and cosine functions to model applied problems

37. 1 (1, 0) (0, 1) (0, -1) (-1, 0)

38. 1 (1, 0) (0, 1) (0, -1) (-1, 0)

39. 1

40. 1

41. When the angle is multiplied, the period is changed. 1

42. When the angle is added to or subtracted from a number, the graph is shifted to the left or right. This is called a phase shift 1

43. When the entire functions is added to or subtracted from a number, the graph is shifted up or down. This is called a vertical translation 1

44. Review for Sine & Cosine Functions  period = 2  for f(x) = sin x & f(x) = cos x * Domain = (– ,  ) Range = [–1, 1] y = asin(k(x – b))+ c OR y = acos(k(x – b))+ c – Amplitude (half the distance between the max and the minimum of the function. Or the distance above or below the midline) = |a| –Period = –Phase Shift (Lt or Rt) = b –Graphing Interval = Range: –Vertical Shift (Up or Dn)= c Factoring may be necessary to get it in this form!!!

45. Amplitude = 3 1 Set graphing interval first Sketch the graph. State the amplitude & period. Divide period into four parts and plot the points accordingly

46. Sketch the graph. State the amplitude & period. Set graphing interval first 1 Amplitude = 1 Divide period into four parts and plot the points accordingly

47. y = a tan k (x – b) y = a cot k (x – b) Tangent & Cotangent Functions –Period = –Phase Shift (Lt or Rt) = b – Graphing Interval = [b, b + ] Tangent has a zero at the beginning, zero at the end, and an asymptote in the middle. Cotangent has an asymptote at the beginning and end, and a zero in the middle.

48. Tangent has a zero at the beginning, zero at the end, and an asymptote in the middle.

49. Cotangent has an asymptote at the beginning and end, and a zero in the middle.

50. y = a sec k (x – b) + c y = a csc k (x – b) + c Secant & Cosecant Functions –Amplitude (vertical distance between U’s and midline) = –Period = –Phase Shift (Lt or Rt) = b – Graphing Interval = [b, b + ] –C is the midline

51. 1 Since it is the reciprocal, where the sine function was equal to zero there is now an asymptote for cosecant function at 0,180,360 degrees.

52. 1 Similarly, where the cosine function was equal to zero there is now an asymptote for the secant function at 90,270 degrees.

53. Key Chapter points: Radian and degree measure Unit Circle Right Triangles Trigonometric functions Graphs of sine and cosine Graphs of other Trigonometric functions Inverse Trigonometric functions