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Precalculus > Final Exam Review > Flashcards

Final Exam Review Flashcards

1
Q

What is sin(theta) also equal to?

A

1/csc(theta)

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2
Q

What is cos(theta) also equal to?

A

1/sec(theta)

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3
Q

What is tan(theta) also equal to?

A

1/cot(theta)

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4
Q

What is csc(theta) also equal to?

A

1/sin(theta)

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5
Q

what is sec(theta) also equal to?

A

1/cos(theta)

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6
Q

what is cot(theta) also equal to?

A

1/tan(theta)

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7
Q

What is the fundamental Pythagorean identity?

A

sin^2(x)+cos^2(x)=1

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8
Q

How can you express cos^2(x) in terms of sin^2(x)?

A

cos^2(x) = 1- sin^2(x)

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9
Q

How can you express sin^2(x) in terms of cos^2(x)?

A

sin^2(x) = 1- cos^2(x)

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10
Q

How can you express sec^2(x) in terms of tan^2(x)?

A

sec^2(x) = 1+ tan^2(x)

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11
Q

How can you express csc^2(x) in terms of cot^2(x)?

A

csc^2(x) = 1+ cot^2(x)

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12
Q

How can you express tan^2(x) in terms of sec^2(x)?

A

tan^2(x)= sec^2(x) - 1

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13
Q

How can you express 1 in terms of tan^2(x) and sec^2(x)?

A

1= sec^2(x) - tan^2(x)

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14
Q

How can you express cot^2(x) in terms of csc^2(x)?

A

cot^2(x)= csc^2(x) -1

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15
Q

How can you express 1 in terms of csc^2(x) and cot^2(x)?

A

1= csc^2(x) - cot^2(x)

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16
Q

What is the formula for converting from radians to degrees?

A

Radians = 180/pi degrees

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17
Q

What is the formula for converting from degrees to radians?

A

Degrees = pi/180

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18
Q

What is the formula for arc length?

A

S = r*theta (always remember to convert to radians)

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19
Q

How do you find the linear velocity of a circle?

A

You set velocity = to r*w and convert w with 2pi then balance the answer to the desired unit.

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20
Q

What is the even identity for cosine?

A

cos(−x)= cos(x)

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21
Q

What is the odd identity for sine?

A

sin(−x)= −sin(x)

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22
Q

What is the odd identity for tangent?

A

tan(−x)= −tan(x)

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23
Q

What is the even identity for secant?

A

sec(−x)= sec(x)

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24
Q

What is the odd identity for cosecant?

A

csc(−x)= −csc(x)

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25
Q

What is the sum identity for sine?

A

sin(A+B) =sin(A)cos(B)+cos(A)sin(B)

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26
Q

What is the odd identity for cotangent?

A

cot(−x)= −cot(x)

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27
Q

What is the sum identity for tangent?

A

tan(A+B) =tan(A)+tan(B)/1−tan(A)tan(B)

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28
Q

What is the difference identity for sine?

A

sin(A−B) =sin(A)cos(B)−cos(A)sin(B)

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29
Q

What is the sum identity for cosine?

A

cos(A+B) =cos(A)cos(B)−sin(A)sin(B)

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30
Q

What is the difference identity for cosine?

A

cos(A−B) =cos(A)cos(B)+sin(A)sin(B)

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31
Q

What is the difference identity for tangent?

A

tan(A−B) =tan(A)-tan(B)/1+tan(A)tan(B)

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32
Q

What is the double-angle identity for sine?

A

sin(2x) =2sin(x)cos(x)

33
Q

What is the double-angle identity for cosine?

A

cos(2x) =cos^2(x)-sin^2(x), cos(2x) =2cos^2(x)-1, 1-2sin^2(x)

34
Q

What is the double-angle identity for tangent?

A

tan(2x) =2tan(x)/1−tan^2(x)

35
Q

What is the power-reduction identity for sin^2(x)?

A

sin^2(x)= 1−cos(2x)/2​

36
Q

What is the power-reduction identity for cos^2(x)?

A

cos^2(x) =1+cos(2x)/2

37
Q

What is the power-reduction identity for tan^2(x)?

A

1−cos(2x)/1+cos(2x)

38
Q

What is the domain and range of sin(x)?

A

D: (-infinity,infinity) R:[-1,1]

39
Q

What is the domain and range of cos(x)?

A

D:(-infinity,infinity) R:[-1,1]

40
Q

What is the domain and range of tan(x)?

A

D:{t|t=\ pi/2 + pik} R:(-infinity,infinity)

41
Q

What is the domain and range of csc(x)?

A

D:{t|t=\ pik, k any integer} R:(-infinity,-1]U[1,infinity)

42
Q

What is the domain and range of sec(x)?

A

D:{t=\ pi/2+pik, k is any integer} R:(-infinity,-1]U[1,infinity)

43
Q

What is the domain and range of cot(x)?

A

D:{t=\ kpi, k is any integer} R:(-infinity,inifinity)

44
Q

What is the domain and range of arcsin(x)?

A

D:[-1,1] R:[-pi/2,pi/2]

45
Q

What is the domain and range of arccos(x)?

A

D:[-1,1] R:[0,pi]

46
Q

What is the domain and range of arctan(x)?

A

D:(-infinity,infinity) R:(-pi/2,pi/2)

47
Q

What is the domain and range of arccsc(x)?

A

D:(-inifinity,-1]U[1,infinity) R:[-pi/2,pi/2],y=\0

48
Q

What is the domain and range of arcsec(x)?

A

D:(-infinity,-1]U[1,infinity) R:[0,pi],y=\pi/2

49
Q

What is the domain and range of arccot(x)?

A

D:(-infinity,infinity) R:(0,pi)

50
Q

What is the formula for difference of Squares?

A

a^2 − b^2 = ( a + b ) ( a − b )

50
Q

How do you factor perfect square trinomials?

A

Check if its a^2+ 2ab + b^2 (or) a^2−2ab + b^2. If the middle term is twice the product of the first and the third term. When true the answer is (a+b)^2

51
Q

What is the quadratic Formula?

A

-b+-sqrt(b^2-4ac)/2a

52
Q

What is the distance formula?

A

d=√((x_2-x_1)²+(y_2-y_1)²)

53
Q

What is the midpoint formula?

A

((x_1 + x_2)/2, (y_1 + y_2)/2)

54
Q

What is the formula for the slope of a line between two points?

A

m = (y_2 - y_1) / (x_2 - x_1)

55
Q

What are the forms for equations of a line?

A

slope-intercept form (y = mx + b), point-slope form (y - y_1) = m(x - x_1)), standard form (Ax + By = C)

56
Q

How do you find the equation for a line parallel to a given line?

A

Know that the two lines have the same slope.

57
Q

How do you find the equation for a line perpendicular to a given line?

A

Know they have opposite reciprocal slopes.

58
Q

What is the formula for average rate of change?

A

(f(b) - f(a)) / (b - a)

59
Q

What are the forms of a quadratic function?

A

standard form (f(x) = ax^2 + bx + c), vertex form (f(x) = a(x - h)^2 + k)

60
Q

How do you find the vertex of a parabola?

A

Using the formula -b/2a to find the axis of symmetry then plugging that x value in to find y

61
Q

How do you find the axis of symmetry of a parabola?

A

Using the formula -b/2a

62
Q

how do you determine the end behavior of a polynomial function?

A

The highest degree of polynomial equations determine the end behavior. – If the degree is even then the ends will extend in the same direction. – If the degree is odd, then the ends will move in opposite directions.

63
Q

What is the for formula for computing the Difference Quotient of a function?

A

(f(x + h) - f(x)) / h

64
Q

Logs – Product property

A

The log of a product is the sum of the logs of the factors:
loga(M⋅N)=logaM+logaN

65
Q

Logs – Quotient property

A

The log of a quotient is the difference between the logs of the numerator and denominator:
logaMN=logaM−logaN

66
Q

Logs – Power property

A

The log of a power is the power times the log of the base:
logaMp=plogaM

67
Q

Logs – Change-of-base formula

A

log_{a}(c)=log_{b}(c)/log_{b}(a)

68
Q

Logs – Equality rule

A

If two logarithms with the same base are equivalent, then what is inside the logarithms are equivalent to each other

69
Q

Logs – Inverse properties

A

for any base “b” where b > 0 and b ≠ 1, logb(b^x) = x and b^(logb(x)) = x; essentially, applying a logarithm to an exponential with the same base results in the original exponent, and vice versa.

70
Q

Logs – Reciprocal property

A

the logarithm of the reciprocal of a number is equal to the negative of the logarithm of that number, mathematically expressed as: logb(1/x) = -logb(x); essentially, taking the logarithm of a reciprocal flips the sign of the result

71
Q

Logs – Other properties

A

log 1 = 0 and logₐ a = 1

72
Q

Formulas for law of cosines

A

a^2 = b^2 + c^2 - 2bc·cosA. b^2 = c^2 + a^2 - 2ca·cosB. c^2 = a^2 + b^2 - 2ab·cosC

73
Q

Area of a triangle for SAS case

A

Area = (1/2) * side1 * side2 * sin(included angle)

74
Q

Law of Cosines if given only sides

A

cos(C) = (a^2 + b^2 - c^2) / (2ab)

75
Q

formula for the magnitude of a vector

A

|v| = √(x² + y²)

76
Q

formula for the direction angle of a vector

A

θ = arctan(y/x)

77
Q

formula for dot product of two vectors

A

a · b = |a| |b| cos(θ)

78
Q

formula for the angle between two vectors using dot product

A

θ = arccos((A · B) / (||A|| ||B||))

Precalculus (2 decks)

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