Unit Circle. Return to. Contents

Transcription

1 Unit Circle Return to Table of Contents 32

2 The Unit Circle The circle x 2 + y 2 = 1, with center (0,0) and radius 1, is called the unit circle. Quadrant II: x is negative and y is positive (0,1) 1 Quadrant I: x and y are both positive (-1,0) Quadrant III: x and y are both negative (0,-1) (1,0) Quadrant IV: x is positive and y is negative 33

3 The Unit Circle The unit circle allows us to extend trigonometry beyond angles of triangles to angles of all measures. (0,1) 1 (a,b) In this triangle, sinθ = b = b 1 (-1,0) θ a b (1,0) cosθ = a 1 = a so the coordinates of (a,b) are also (cosθ, sinθ) (0,-1) For any angle in standard position, the point where the terminal side of the angle intercepts the circle is called the terminal point. 34

4 Terminal Point In this example, the terminal point is in Quadrant IV If we look at the triangle, we can see that sin(-55 ) = 0.82 cos(-55 ) = 0.57 EXCEPT that we have to take the direction into account, and so sin(-55 ) is negative because the y value is below the x-axis. For any angle θ in standard position, the terminal point has coordinates (cosθ, sinθ). 35

5 Unit Circle Click on this text to go to the Khan Academy Unit Circle Manipulative try some problems: 36

6 Finding Coordinates of a Point What are the coordinates of point C? In this example, we know the angle. Using a calculator, we find that cos and sin , so the coordinates of C are approximately (0.72, 0.69). 1 Note that ! 37

7 The Tangent Function Recall SOH-CAH-TOA sin = θ cos θ= tan θ= opp hyp adj hyp opp adj opposite side hypotenuse θ adjacent side Why? It is also true that sin tan =. opp hyp adj hyp opp hyp hyp adj θ cos θ θ = = opp = tan adj θ 38

8 Angles in the Unit Circle Example: Given a terminal point on the unit circle ( ). Find the value of cos, sin and tan of the angle. Solution: Let the angle be. x = cos, so cos =. y = sin, so sin =. tan = = = = (Shortcut: Just cross out the 41's in the complex fraction.) 39

9 Finding Cos, Sin, and Tan Example: Given a terminal point, find θ, tanθ and cscθ. To find θ, use sin -1 or cos -1 : sin -1 ( ) = θ θ 28.1 tan θ = sin θ/ cos θ tan θ = csc θ = 1/ sin θ csc θ = Note the "hidden" Pythagorean Triple, 8, 15, 17). 40

10 Finding Cos, Sin, and Tan Example: Find the x-value of point A, θ and the tan θ. For every point on the circle, θ A (, - 13 ) Since x is in quadrant III, x = (- 5 ) sin , BUT θ is in quadrant III, so θ = = (notice how and have the same sine) tan θ = sin θ cos θ = =

11 Finding Cos, Sin, and Tan Example: Given the terminal point of ( -5 /13, -12 /13). Find sin x, cos x, and tan x. Answer 42

12 14 What is tan θ? 3 (- 5, ) A θ B C Answer D 43

13 15 What is sin θ? 3 (- 5, ) A θ B C Answer D 44

14 16 What is θ (give your answer to the nearest degree)? 3 (- 5, ) θ Answer 45

15 17 Given the terminal point, find tan x. Answer 46

16 18 Knowing tan x = Find sin x if the terminal point is in the 2 nd quadrant Answer 47

17 Equilateral and Isosceles Triangles Equilateral and isosceles triangles occur frequently in geometry and trigonometry. The angles in these triangles are multiples of 30 and 45. A calculator will give approximate values for the trig functions of these angles, but we often need to know the exact values. Isosceles Right Triangle Equilateral Triangle (the altitude divides the triangle into two triangles) 48

18 Special Right Triangles (see Triangle Trig Review unit for more detail on this topic) 49

19 Special Triangles and the Unit Circle (-, ) Multiples of 45 angles have sin and cos of ±, depending on the quadrant. 50

20 Unit Circle 51

21 Unit Circle Drag the degree and radian angle measures to the angles of the circle: π π 4 2π 5π 4 π 2 3π 2 3π 4 7π 4 52

22 Fill in the coordinates of x and y for each point on the unit circle: 3π 4 Unit Circle π 2 π 4 π 2π π 4 3π 2 7π

23 Special Triangles and the Unit Circle Angles that are multiples of 30 have sin and cos of ± and ±. 54

24 π 6 π 2 π π 6 0 2π 4π 3 π 3 3π 2 2π 3 5π 3 Unit Circle Drag the degree and radian angle measures to the angles of the circle: 7π 6 11π 6 55

25 Unit Circle Drag in the coordinates of x and y for each point on the unit circle: π 5π 6 2π 3 7π 6 4π 3 π 2 3π 2 5π 3 π 3 11π 6 π 6 2π

26 Special Angles in Degrees 57

27 Radian Values of Special Angles 58

28 Exact Values of Special Angles 59

29 Put it all together... 60

30 Exact Values of Special Angles Complete the table below: Degrees Radians sin θ cos θ tan θ 61

31 Exact Values of Special Angles 62

32 Exact Values of Special Angles 1 63

33 Finding Angles and Trig Values If we know one trig function value and the quadrant in which the angle lies, we can find the angle and the other trig values. 64

34 Finding Angles and Trig Values Example: If tan =, and sin < 0, find sin, cos and the value of. Solution: Since tan is positive and sin is negative, the terminal side of Draw a right triangle in Quadrant III. Use the Pythagorean Theorem to find the length of the hypotenuse: must be in Quadrant III. opp -3 adj hyp (Continued on next slide) 65

35 Finding Angles and Trig Values Once we know the lengths for each side, we can calculate the sin, cos and the angle. Used the signed numbers to get the correct values. sin = = opp -3 adj hyp cos = = Use any inverse trig function to find the angle. tan-1( ) Because the angle is in QIII, we need to add = 216.7, so

36 19 A B C D E F G H I J Answer 67

37 20 A B C D E F G H I J Answer 68

38 21 A B C D E F G H I J Answer 69

39 22 Which functions are positive in the second quadrant? Choose all that apply. A B C D E F cos x sin x tan x sec x csc x cot x Answer 70

40 23 Which functions are positive in the fourth quadrant? Choose all that apply. A B C D E F cos x sin x tan x sec x csc x cot x Answer 71

41 24 Which functions are positive in the third quadrant? Choose all that apply. A B C D E F cos x sin x tan x sec x csc x cot x Answer 72