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Calculus II > Trigonometric Identities > Flashcards

Trigonometric Identities Flashcards

1
Q

which function does sin have a reciprocal relationship with?

A

csc

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2
Q

which function does cos have a reciprocal relationship with?

A

sec

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3
Q

which function does tan have a reciprocal relationship with?

A

cot

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4
Q

which function does csc have a reciprocal relationship with?

A

sin

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5
Q

which function does sec have a reciprocal relationship with?

A

cos

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6
Q

which function does cot have a reciprocal relationship with?

A

tan

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7
Q

power reduction of sin^2(u)

A

1/2(1-cos(2u))

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8
Q

power reduction of cos^2(u)

A

1/2(1+cos(2u))

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9
Q

power reduction of tan^2(u)

A

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

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10
Q

double angle identity: sin(2u)

A

2sinucosu

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11
Q

double angle identity: cos(2u)

A

cos^2(u) - sin^2(u)

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12
Q

pythagorean identity: tan^2(u)

A

sec^2(u) - 1

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13
Q

pythagorean identity: sec^2(u)

A

tan^2(u) + 1

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14
Q

substitution: sqrt(a^2 + u^2)

A

u = (a)(tan(theta))

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15
Q

substitution: sqrt(a^2 - u^2)

A

u = (a)(sin(theta))

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16
Q

substitution: sqrt(u^2 - a^2)

A

u = (a)(sec(theta))

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17
Q

arc length s of curve f(x)

A

s = definite integral from a to b of sqrt(1 + (f’(x))^2) dx

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18
Q

Trap (n) = , also delta x =

A

(1/2)(delta x)[f(x,0) + 2f(x,1) + 2f(x,2) + … … + 2f(x,n-1) + f(x,n)]

delta x = (upper bound of integral - lower bound of integral)/(number of sub intervals ie terms in series)

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19
Q

Mid (n) = , also delta x =

A

(delta x)[f(x,1) + f(x,2) … … + f(x,n)]

delta x = (upper bound of integral - lower bound of integral)/(number of sub intervals ie terms in series)

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20
Q

Simp (n) = , also delta x =

A

(1/3)(delta x)[f(x,0) + 4f(x,1) + 2f(x,2) + 4f(x,3) + 2f(x,4) … … 4f(x,n-1) + f(x,n)]

n must be an even number.

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21
Q

separable differential equation

A

dy/dx = (f(y))(g(x))

when derivative is isolated the other side of the equation can be factored so that one factor is a function of only y and the other factor is a function of only x.

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22
Q

define implicit form of a differential equation

A

implicit form is not solved for y in terms of x (y is not completely isolated on one side)

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23
Q

define explicit form of a differential equation

A

explicit form is solved for y in terms of x (y is completely isolated on one side)

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24
Q

Integral Identity: integral of u^n du , when n does not equal 1

A

u^(n+1) / n+1 + C

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25
Q

Integral Identity: integral of u^-1 du

A

ln|u| + C

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26
Q

Integral Identity: integral of e^u

A

e^u + C

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27
Q

Integral Identity: integral of a^u , when a does not equal 1

A

(1 / (lna) ) (a^u) + C

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28
Q

Trig Integral Identity: integral of cos(u)

A

sin(u) + C

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29
Q

Trig Integral Identity: integral of sin(u)

A

-cos(u) + C

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30
Q

Trig Integral Identity: integral of (sec u)(tan u) du

“a sea can tan if a sea can can.”

A

sec(u) + C

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31
Q

Trig Integral Identity: integral of sec^2(u) du

“A sea can square its t….”

A

tan(u) + C

“A sea can square its toes”

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32
Q

Trig Integral Identity: integral of (csc u)(cot u) du

“a cosy cot can if a cosy can can’t.”

A

-csc(u) + C

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33
Q

Trig Integral Identity: integral of csc^2(u) du

“a cozy can square without a cot there.”

A

-cot(u) + C

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34
Q

Trig Integral Identity: integral of tan(u) du

“that tan has no lines cuz!”

A

-ln| cos(u) | + C

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35
Q

Trig Integral Identity: integral of cot(u) du

“that cot lines its signs!”

A

ln| sin(u) | + C

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36
Q

Trig Integral Identity: integral of sec(u) du

“A sea can always lines it’s sea cans with tans.”

A

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

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37
Q

Trig Integral Identity: integral of csc(u) du

“A cosy can alone can’t line it’s coats with cosy cans.”

A

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

38
Q

Trig Integral Identity: integral of 1 / sqrt(a^2 - u^2) du

A

sin^-1 (u/a) + C

39
Q

Trig Integral Identity: integral of 1 / sqrt(a^2 + u^2) du

A

(1/a)(tan^-1 (u/a)) + C

40
Q

Integral Identity: integral of ln(u)

A

(x)(ln(x)) - x + C

41
Q

Integral Identity: integral of (u)(dv) , ie two statements multiplied

A

(u)(v) - integral of (v)(du)

42
Q

Trig Integral Identity: integral of 1 / ((u)(sqrt(u^2 - a^2) ) )

A

(1/a)(sec^-1( | u/a | ) + C

43
Q

Derivative of sin^-1 (u)

A

u’ / sqrt(1 - u^2 )

44
Q

Derivative of tan^-1 (u)

A

u’ / (1 + u^2 )

45
Q

Derivative of sec^-1 (u)

A

(u’) / ( |u| )(sqrt(u^2 - 1) )

46
Q

Pythagorean Identity of cot^2(theta)

A

csc^2(theta) - 1

47
Q

Pythagorean Identity of csc^2(theta)

A

cot^2(theta) + 1

48
Q

circle identity of sin(theta)

A

y/r

49
Q

circle identity of cos(theta)

A

x/r

50
Q

circle identity of tan(theta)

A

y/x

51
Q

circle identity of csc(theta)

A

r/y

52
Q

circle identity of sec(theta)

A

r/x

53
Q

circle identity of cot(theta)

A

x/y

54
Q

cofunction identity (ie pi/2 - theta) of sin(theta)

A

cos(pi/2 - theta)

55
Q

cofunction identity (ie pi/2 - theta) of cos(theta)

A

sin(pi/2-theta)

56
Q

cofunction identity (ie pi/2 - theta) of tan(theta)

A

cot(pi/2-theta)

57
Q

cofunction identity (ie pi/2 - theta) of cot(theta)

A

tan(pi/2-theta)

58
Q

cofunction identity (ie pi/2 - theta) of sec(theta)

A

csc(pi/2-theta)

59
Q

cofunction identity (ie pi/2 - theta) of csc(theta)

A

sec(pi/2-theta)

60
Q

what does sin(theta) equal in the unit circle

A

y

61
Q

what does cos(theta) equal in the unit circle

A

x

62
Q

what does tan(theta) equal in the unit circle

A

y/x

63
Q

even odd property of sin(-theta)

A

-sin(theta)

64
Q

even odd property of -sin(theta)

A

sin(-theta)

65
Q

even odd property of cos(-theta)

A

cos(theta)

66
Q

even odd property of cos(theta)

A

cos(-theta)

67
Q

even odd property of -cos(theta)

A

-cos(theta) ie unchanged

68
Q

even odd property of tan(-theta)

A

-tan(theta)

69
Q

even odd property of -tan(theta)

A

tan(-theta)

70
Q

even odd property of csc(-theta)

A

-csc(theta)

71
Q

even odd property of sec(-theta)

A

sec(theta)

72
Q

even odd property of sec(theta)

A

sec(-theta)

73
Q

even odd property of cot(-theta)

A

-cot(theta)

74
Q

even odd property of -cot(theta)

A

cot(-theta)

75
Q

how do you find sin/cos/tan etc of (pi/x)?

A

draw a triangle with the angles (45x45x90, 30x60x90), label with unit circle values, solve pyth theorem, use soh cah toa.

76
Q

sin’(x) =

A

x’cos(x) + C

77
Q

cos’(x) =

A

-x’sin(x) + C

78
Q

tan’(x) =

A

x’sec^2(x) + C

79
Q

sec’(x) =

A

x’sec(x)tan(x) + C

80
Q

csc’(x) =

A

-x’csc(x)cot(x) + C

81
Q

cot’(x) =

A

-x’csc^2(x) + C

82
Q

if f(x) is position what is f’(x)

A

velocity

83
Q

if f(x) is position what is f’‘(x)

A

acceleration

84
Q

if f(x) is position what is | f’(x) |

A

speed

85
Q

arcsin’(x) =

A

1/sqrt(1-x^2) + C

86
Q

arccos’(x) =

A

-1/sqrt(1-x^2) + C

87
Q

arctan’(x) =

A

1/(1+x^2) + C

88
Q

arccsc’(x) =

A

-1/(( |x| )(sqrt(x^2 - 1)) + C

89
Q

arcsec’(x) =

A

1/(( |x| )(sqrt(x^2 - 1)) + C

90
Q

arccot’(x) =

A

-1/(1 + x^2)

Calculus II (1 decks)

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