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Math 113 > Trig Identities > Flashcards

Trig Identities Flashcards

1
Q

sin²(x)

A

1 - cos²(x)

1/2[1 - cos(2x)]

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2
Q

cos²(x)

A

1 - sin²(x)

1/2[1 + cos(2x)]

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3
Q

sec²(x)

A

1 + tan²(x)

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4
Q

tan²(x)

A

sec²(x) - 1

1 - cos(2x))/(1 + cos(2x)

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5
Q

csc²(x)

A

1 + cot²(x)

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6
Q

cot²(x)

A

csc²(x) - 1

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7
Q

Pythagorean Identities

A

sin²(x) + cos²(x) = 1
1 + tan²(x) = sec²(x)
1 + cot²(x) = csc²(x)

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8
Q

sin(π/2 - x)

A

cos(x)

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9
Q

csc(π/2 - x)

A

sec(x)

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10
Q

sec(π/2 - x)

A

csc(x)

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11
Q

cos(π/2 - x)

A

sin(x)

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12
Q

tan(π/2 - x)

A

cot(x)

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13
Q

cot(π/2 - x)

A

tan(x)

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14
Q

Cofunction Identities

A 
sin(π/2 - x) = cos(x)
cos(π/2 - x) = sin(x)
tan(π/2 - x) = cot(x)
csc(π/2 - x) = sec(x)
sec(π/2 - x) = csc(x)
cot(π/2 - x) = tan(x)
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15
Q

sin(a + b)

A

sin(a)cos(b) + sin(b)cos(a)

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16
Q

sin(a - b)

A

sin(a)cos(b) - sin(b)cos(a)

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17
Q

cos(a + b)

A

cos(a)cos(b) - sin(a)sin(b)

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18
Q

cos(a - b)

A

cos(a)cos(b) + sin(a)sin(b)

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19
Q

tan(a + b)

A

(tan(a) + tan(b))/(1 - tan(a)tan(b))

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20
Q

tan(a - b)

A

(tan(a) - tan(b))/(1 + tan(a)tan(b))

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21
Q

Sum and Difference Formulas

A

sin(a + b) = sin(a)cos(b) + sin(b)cos(a)
sin(a - b) = sin(a)cos(b) - sin(b)cos(a)
cos(a + b) = cos(a)cos(b) - sin(a)sin(b)
cos(a - b) = cos(a)cos(b) + sin(a)sin(b)
tan(a + b) = (tan(a) + tan(b))/(1 - tan(a)tan(b))
tan(a - b) = (tan(a) - tan(b))/(1 + tan(a)tan(b))

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22
Q

sin(2x)

A

2sin(x)cos(x)

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23
Q

cos(2x)

A

cos²(x) - sin²(x)
2cos²(x) - 1
1 - 2sin²(x)

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24
Q

tan(2x)

A

(2tan(x))/(1 - tan²x)

25
Q

Double Angle Formulas

A 
sin(2x) = 2sin(x)cos(x)
cos(2x) = cos²(x) - sin²(x) 
cos(2x) = 2cos²(x) - 1
cos(2x) = 1 - 2sin²(x)
tan(2x) = (2tan(x))/(1 - tan²x)
26
Q

sin(a) + sin(b)

A

2sin((a+b)/2)cos((a-b)/2)

27
Q

sin(a) - sin(b)

A

2cos((a+b)/2)sin((a-b)/2)

28
Q

cos(a) + cos(b)

A

2cos((a+b)/2)cos((a-b)/2)

29
Q

cos(a) - cos(b)

A

-2sin((a+b)/2)sin((a-b)/2)

30
Q

Sum-to-Product Formulas

A

sin(a) + sin(b) = 2sin((a+b)/2)cos((a-b)/2)
sin(a) - sin(b) = 2cos((a+b)/2)sin((a-b)/2)
cos(a) + cos(b) = 2cos((a+b)/2)cos((a-b)/2)
cos(a) - cos(b) = -2sin((a+b)/2)sin((a-b)/2)

31
Q

sin(a)sin(b)

A

1/2[cos(a - b) - cos(a + b)]

32
Q

cos(a)cos(b)

A

1/2[cos(a - b) + cos(a + b)]

33
Q

sin(a)cos(b)

A

1/2[sin(a + b) + sin(a - b)]

34
Q

cos(a)sin(b)

A

1/2[sin(a + b) - sin(a - b)]

35
Q

Product to Sum Formulas

A 
sin(a)sin(b) = 1/2[cos(a - b) - cos(a + b)]
cos(a)cos(b) = 1/2[cos(a - b) + cos(a + b)]
sin(a)cos(b) = 1/2[sin(a + b) + sin(a - b)]
cos(a)sin(b) = 1/2[sin(a + b) - sin(a - b)]
36
Q

Circular Function Definitions

A 
0 < Ɵ < π/2
sin(Ɵ) = y/r
cos(Ɵ) = x/r
tan(Ɵ) = y/x
csc(Ɵ) = r/y
sec(Ɵ) = r/x
cot(Ɵ) = x/y
37
Q

Right Triangle Definitions

A 
sin(Ɵ) = opp/hyp
cos(Ɵ) = adj/hyp
tan(Ɵ) = opp/adj
csc(Ɵ) = hyp/opp
sec(Ɵ) = hyp/adj
cot(Ɵ) = adj/opp
38
Q

√(a² - b²x²)

A

x = (a/b)sin(Ɵ)

1 - sin²(Ɵ) = cos²(Ɵ)

39
Q

√(a² + b²x²)

A

x = (a/b)tan(Ɵ)

1 + tan²(Ɵ) = sec²(Ɵ)

40
Q

√(b²x² - a²)

A

x = (a/b)sec(Ɵ)

sec²(Ɵ) -1 = tan²(Ɵ)

41
Q

d/dx(sin(x))

42
Q

d/dx(cos(x))

43
Q

d/dx(tan(x))

44
Q

d/dx(cot(x))

45
Q

d/dx(sec(x))

A

sec(x)tan(x)

46
Q

d/dx(csc(x))

A

-csc(x)cot(x)

47
Q

∫cos(x)dx

A

sin(x) + C

48
Q

∫sin(x)dx

A

-cos(x) + C

49
Q

∫sec²(x)dx

A

tan(x) + C

50
Q

∫csc²(x)dx

A

-cot(x) + C

51
Q

∫sec(x)tan(x)dx

A

sec(x) + C

52
Q

∫csc(x)cot(x)dx

A

-csc(x) + C

53
Q

∫tan(x)dx

A

-ln|cos(x)| + C

54
Q

∫cot(x)dx

A

ln|sin(x)| + C

55
Q

∫sec(x)dx

A

ln|sec(x) + tan(x)| + C

56
Q

∫csc(x)dx

A

ln|csc(x) - cot(x)| + C

57
Q

Half Angle Formulas

A 
sin²(x) = [1 - cos(2x)]/2
cos²(x) = [1 + cos(2x)]/2
tan²(x) = [1 - cos(2x)]/[1 + cos(2x)]

sin(x/2) = ± √[(1 - cos(x))/2)]
cos(x/2) = ± √[1 + cos(x)/2)]
tan(x/2) = ± √[(1 - cos(x))/(1 + cos(x))]
tan(x/2) = (1 - cos(x))/(sin(x))
tan(x/2) = (sin(x))/(1 + cos(x))
58
Q

sin²(x)cos²(x)

A

[1/2sin(2x)]²

Math 113 (3 decks)

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