Trig review. If θ is measured counterclockwise from the positive x axis and (x,y) is on the unit circle, we define sin and cos so that

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1 Today

2 If is measured counterclockwise from the positive x axis and (x,y) is on the unit circle, we define sin and cos so that (A) x=sin(), y=tan(). (B) x=tan(), y=sin(). (C) x=sin(), y=cos(). (D) x=cos(), y=sin().

3 If is measured counterclockwise from the positive x axis and (x,y) is on the unit circle, we define sin and cos so that (A) x=sin(), y=tan(). (B) x=tan(), y=sin(). (C) x=sin(), y=cos(). (D) x=cos(), y=sin().

4 If is measured counterclockwise from the positive x axis and (x,y) is on the unit circle, we define sin and cos so that (A) x=sin(), y=tan(). (B) x=tan(), y=sin(). (C) x=sin(), y=cos(). Because (x,y) is on the unit circle, we know that cos 2 () + sin 2 () = 1. (D) x=cos(), y=sin().

5 (cos, sin) Learn special angles in Quad I and modify signs for other Quads.

6 (cos, sin) (cos, sin) Learn special angles in Quad I and modify signs for other Quads.

7 (cos, sin) (cos, sin) Learn special angles in Quad I and modify signs for other Quads.

8 (cos, sin) (cos, sin) Learn special angles in Quad I and modify signs for other Quads. cos = -cos

9 (cos, sin) (cos, sin) Learn special angles in Quad I and modify signs for other Quads. cos = -cos sin = sin

10 (cos, sin) (cos, sin) Learn special angles in Quad I and modify signs for other Quads. cos = -cos sin = sin (cos, sin)

11 (cos, sin) (cos, sin) Learn special angles in Quad I and modify signs for other Quads. cos = -cos sin = sin (cos, sin)

12 (cos, sin) (cos, sin) Learn special angles in Quad I and modify signs for other Quads. cos = -cos sin = sin cos = -cos (cos, sin)

13 (cos, sin) (cos, sin) Learn special angles in Quad I and modify signs for other Quads. cos = -cos sin = sin cos = -cos sin = -sin (cos, sin)

14 (cos, sin) (cos, sin) Learn special angles in Quad I and modify signs for other Quads. cos = -cos sin = sin cos = -cos sin = -sin (cos, sin) (cos, sin)

15 (cos, sin) (cos, sin) Learn special angles in Quad I and modify signs for other Quads. cos = -cos sin = sin cos = -cos sin = -sin (cos, sin) (cos, sin)

16 (cos, sin) (cos, sin) Learn special angles in Quad I and modify signs for other Quads. cos = -cos sin = sin cos = -cos sin = -sin (cos, sin) cos = cos (cos, sin)

17 (cos, sin) (cos, sin) Learn special angles in Quad I and modify signs for other Quads. cos = -cos sin = sin cos = -cos sin = -sin (cos, sin) cos = cos sin = -sin (cos, sin)

18 The other trig functions:

19 The other trig functions: tan = sin / cos

20 The other trig functions: tan = sin / cos csc = 1 / sin

21 The other trig functions: tan = sin / cos csc = 1 / sin sec = 1 / cos

22 The other trig functions: tan = sin / cos csc = 1 / sin sec = 1 / cos cot = 1 / tan

23 Which of the following is not a trig identity? (A) 1 + cot 2 = csc 2 (B) tan = sec 2 (C) sin(2) = 2 sin cos (D) cos() = sin(-π/2) (E) sin() = cos(-π/2)

24 Which of the following is not a trig identity? (A) 1 + cot 2 = csc 2 (B) tan = sec 2 (C) sin(2) = 2 sin cos (D) cos() = sin(-π/2) (E) sin() = cos(-π/2) cos(a+b) = cosa cosb - sina sinb

25 Which of the following is not a trig identity? (A) 1 + cot 2 = csc 2 (B) tan = sec 2 (C) sin(2) = 2 sin cos (D) cos() = sin(-π/2) (E) sin() = cos(-π/2) cos(a+b) = cosa cosb - sina sinb 1 cos sin

26 Which of the following is not a trig identity? (A) 1 + cot 2 = csc 2 sin 2 + cos 2 = 1 (B) tan = sec 2 (C) sin(2) = 2 sin cos (D) cos() = sin(-π/2) (E) sin() = cos(-π/2) cos(a+b) = cosa cosb - sina sinb 1 cos sin

27 Which of the following is not a trig identity? (A) 1 + cot 2 = csc 2 (B) tan = sec 2 sin 2 + cos 2 = 1 sin 2 sin 2 sin 2 (C) sin(2) = 2 sin cos (D) cos() = sin(-π/2) (E) sin() = cos(-π/2) cos(a+b) = cosa cosb - sina sinb 1 cos sin

28 Which of the following is not a trig identity? (A) 1 + cot 2 = csc 2 (B) tan = sec 2 sin 2 + cos 2 = 1 cos 2 cos 2 cos 2 (C) sin(2) = 2 sin cos (D) cos() = sin(-π/2) (E) sin() = cos(-π/2) cos(a+b) = cosa cosb - sina sinb 1 cos sin

29 Which of the following is not a trig identity? (A) 1 + cot 2 = csc 2 (B) tan = sec 2 (C) sin(2) = 2 sin cos (D) cos() = sin(-π/2) (E) sin() = cos(-π/2) sin 2 + cos 2 = 1 cos 2 cos 2 cos 2 <-- Use sin(a+b) (watch today s 2 nd video) 1 sin cos(a+b) = cosa cosb - sina sinb cos

30 Which of the following is not a trig identity? (A) 1 + cot 2 = csc 2 (B) tan = sec 2 sin 2 + cos 2 = 1 cos 2 cos 2 cos 2 (C) sin(2) = 2 sin cos (D) cos() = sin(-π/2) (E) sin() = cos(-π/2) <-- Use sin(a+b) (watch today s 2 nd video) Know graphs, how to shift or use sin(a+b), cos(a+b) cos(a+b) = cosa cosb - sina sinb 1 cos sin

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32 The SI convention for the units used for angles is radians, not degrees. Although degrees date back thousands of years they are less convenient, for example, in calculating the arclength:

33 The SI convention for the units used for angles is radians, not degrees. Although degrees date back thousands of years they are less convenient, for example, in calculating the arclength: Which is nicer: s = r or s = π r / 360?

34 The SI convention for the units used for angles is radians, not degrees. Although degrees date back thousands of years they are less convenient, for example, in calculating the arclength: Which is nicer: s = r or s = π r / 360? Just say no to degrees.

35 The SI convention for the units used for angles is radians, not degrees. Although degrees date back thousands of years they are less convenient, for example, in calculating the arclength: Which is nicer: s = r or s = π r / 360? Just say no to degrees. Unless you re looking at a map.

36 The SI convention for the units used for angles is radians, not degrees. Although degrees date back thousands of years they are less convenient, for example, in calculating the arclength: Which is nicer: s = r or s = π r / 360? Just say no to degrees. Unless you re looking at a map. Or baking. Or trying to find a job.

37 cos(2π/3) = (A) (B) (C) (D) p 3 2 p

38 cos(2π/3) = (A) (B) (C) (D) p 3 2 p π/3

39 cos(2π/3) = (A) (B) (C) (D) p 3 2 p π/3 2π/3

40 cos(2π/3) = (A) (B) (C) (D) p 3 2 p π/3 2π/3 π/3

41 cos(2π/3) = (A) p 3 2 p 3 π/3 2π/3 (B) (C) (D) π/3 1/2 3/2

42 cos(2π/3) = (A) p 3 2 p 3 π/3 2π/3 (B) (C) (D) And 2π/3 is in Quad II so cos(2π/3) < 0. 1 π/3 1/2 3/2

43 tan ( /4) = 1 (A) p 2 (B) 1 p (C) 2 (D) 1 2

44 tan ( /4) = 1 (A) p 2 (B) 1 p (C) 2 (D) 1 2

45 Which of the following is false? (A) cos(arctan(sqrt(3))) = 1/2 (B) sin(arccos(1/2)) = sqrt(3)/2 (C) arctan(1) = π/4 (D) arcsin(1/2) = π/3 (E) sin(3π/2) = -1 Note: cos -1 (x) = arccos(x), tan -1 (x) = arctan(x).

46 Which of the following is false? (A) cos(arctan(sqrt(3))) = 1/2 (B) sin(arccos(1/2)) = sqrt(3)/2 (C) arctan(1) = π/4 (D) arcsin(1/2) = π/3 (E) sin(3π/2) = -1

47 For the more ambitious student:

48 For the more ambitious student:

49 For the more ambitious student:

50 For the more ambitious student:

51 For the more ambitious student:

52 For the more ambitious student:

53 For the more ambitious student:

54 For the more ambitious student:

55 For the more ambitious student:

56 For the more ambitious student:

57 For the more ambitious student:

58 For the more ambitious student: cos(x+y) = cos(x)cos(y) - sin(x)sin(y)

59 For the more ambitious student: sin(x+y) = sin(x)cos(y) + sin(y)cos(x)