Final Exam Review Flashcards
What is sin(theta) also equal to?
1/csc(theta)
What is cos(theta) also equal to?
1/sec(theta)
What is tan(theta) also equal to?
1/cot(theta)
What is csc(theta) also equal to?
1/sin(theta)
what is sec(theta) also equal to?
1/cos(theta)
what is cot(theta) also equal to?
1/tan(theta)
What is the fundamental Pythagorean identity?
sin^2(x)+cos^2(x)=1
How can you express cos^2(x) in terms of sin^2(x)?
cos^2(x) = 1- sin^2(x)
How can you express sin^2(x) in terms of cos^2(x)?
sin^2(x) = 1- cos^2(x)
How can you express sec^2(x) in terms of tan^2(x)?
sec^2(x) = 1+ tan^2(x)
How can you express csc^2(x) in terms of cot^2(x)?
csc^2(x) = 1+ cot^2(x)
How can you express tan^2(x) in terms of sec^2(x)?
tan^2(x)= sec^2(x) - 1
How can you express 1 in terms of tan^2(x) and sec^2(x)?
1= sec^2(x) - tan^2(x)
How can you express cot^2(x) in terms of csc^2(x)?
cot^2(x)= csc^2(x) -1
How can you express 1 in terms of csc^2(x) and cot^2(x)?
1= csc^2(x) - cot^2(x)
What is the formula for converting from radians to degrees?
Radians = 180/pi degrees
What is the formula for converting from degrees to radians?
Degrees = pi/180
What is the formula for arc length?
S = r*theta (always remember to convert to radians)
How do you find the linear velocity of a circle?
You set velocity = to r*w and convert w with 2pi then balance the answer to the desired unit.
What is the even identity for cosine?
cos(−x)= cos(x)
What is the odd identity for sine?
sin(−x)= −sin(x)
What is the odd identity for tangent?
tan(−x)= −tan(x)
What is the even identity for secant?
sec(−x)= sec(x)
What is the odd identity for cosecant?
csc(−x)= −csc(x)
What is the sum identity for sine?
sin(A+B) =sin(A)cos(B)+cos(A)sin(B)
What is the odd identity for cotangent?
cot(−x)= −cot(x)
What is the sum identity for tangent?
tan(A+B) =tan(A)+tan(B)/1−tan(A)tan(B)
What is the difference identity for sine?
sin(A−B) =sin(A)cos(B)−cos(A)sin(B)
What is the sum identity for cosine?
cos(A+B) =cos(A)cos(B)−sin(A)sin(B)
What is the difference identity for cosine?
cos(A−B) =cos(A)cos(B)+sin(A)sin(B)
What is the difference identity for tangent?
tan(A−B) =tan(A)-tan(B)/1+tan(A)tan(B)
What is the double-angle identity for sine?
sin(2x) =2sin(x)cos(x)
What is the double-angle identity for cosine?
cos(2x) =cos^2(x)-sin^2(x), cos(2x) =2cos^2(x)-1, 1-2sin^2(x)
What is the double-angle identity for tangent?
tan(2x) =2tan(x)/1−tan^2(x)
What is the power-reduction identity for sin^2(x)?
sin^2(x)= 1−cos(2x)/2
What is the power-reduction identity for cos^2(x)?
cos^2(x) =1+cos(2x)/2
What is the power-reduction identity for tan^2(x)?
1−cos(2x)/1+cos(2x)
What is the domain and range of sin(x)?
D: (-infinity,infinity) R:[-1,1]
What is the domain and range of cos(x)?
D:(-infinity,infinity) R:[-1,1]
What is the domain and range of tan(x)?
D:{t|t=\ pi/2 + pik} R:(-infinity,infinity)
What is the domain and range of csc(x)?
D:{t|t=\ pik, k any integer} R:(-infinity,-1]U[1,infinity)
What is the domain and range of sec(x)?
D:{t=\ pi/2+pik, k is any integer} R:(-infinity,-1]U[1,infinity)
What is the domain and range of cot(x)?
D:{t=\ kpi, k is any integer} R:(-infinity,inifinity)
What is the domain and range of arcsin(x)?
D:[-1,1] R:[-pi/2,pi/2]
What is the domain and range of arccos(x)?
D:[-1,1] R:[0,pi]
What is the domain and range of arctan(x)?
D:(-infinity,infinity) R:(-pi/2,pi/2)
What is the domain and range of arccsc(x)?
D:(-inifinity,-1]U[1,infinity) R:[-pi/2,pi/2],y=\0
What is the domain and range of arcsec(x)?
D:(-infinity,-1]U[1,infinity) R:[0,pi],y=\pi/2
What is the domain and range of arccot(x)?
D:(-infinity,infinity) R:(0,pi)
What is the formula for difference of Squares?
a^2 − b^2 = ( a + b ) ( a − b )
How do you factor perfect square trinomials?
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
What is the quadratic Formula?
-b+-sqrt(b^2-4ac)/2a
What is the distance formula?
d=√((x_2-x_1)²+(y_2-y_1)²)
What is the midpoint formula?
((x_1 + x_2)/2, (y_1 + y_2)/2)
What is the formula for the slope of a line between two points?
m = (y_2 - y_1) / (x_2 - x_1)
What are the forms for equations of a line?
slope-intercept form (y = mx + b), point-slope form (y - y_1) = m(x - x_1)), standard form (Ax + By = C)
How do you find the equation for a line parallel to a given line?
Know that the two lines have the same slope.
How do you find the equation for a line perpendicular to a given line?
Know they have opposite reciprocal slopes.
What is the formula for average rate of change?
(f(b) - f(a)) / (b - a)
What are the forms of a quadratic function?
standard form (f(x) = ax^2 + bx + c), vertex form (f(x) = a(x - h)^2 + k)
How do you find the vertex of a parabola?
Using the formula -b/2a to find the axis of symmetry then plugging that x value in to find y
How do you find the axis of symmetry of a parabola?
Using the formula -b/2a
how do you determine the end behavior of a polynomial function?
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.
What is the for formula for computing the Difference Quotient of a function?
(f(x + h) - f(x)) / h
Logs – Product property
The log of a product is the sum of the logs of the factors:
loga(M⋅N)=logaM+logaN
Logs – Quotient property
The log of a quotient is the difference between the logs of the numerator and denominator:
logaMN=logaM−logaN
Logs – Power property
The log of a power is the power times the log of the base:
logaMp=plogaM
Logs – Change-of-base formula
log_{a}(c)=log_{b}(c)/log_{b}(a)
Logs – Equality rule
If two logarithms with the same base are equivalent, then what is inside the logarithms are equivalent to each other
Logs – Inverse properties
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.
Logs – Reciprocal property
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
Logs – Other properties
log 1 = 0 and logₐ a = 1
Formulas for law of cosines
a^2 = b^2 + c^2 - 2bc·cosA. b^2 = c^2 + a^2 - 2ca·cosB. c^2 = a^2 + b^2 - 2ab·cosC
Area of a triangle for SAS case
Area = (1/2) * side1 * side2 * sin(included angle)
Law of Cosines if given only sides
cos(C) = (a^2 + b^2 - c^2) / (2ab)
formula for the magnitude of a vector
|v| = √(x² + y²)
formula for the direction angle of a vector
θ = arctan(y/x)
formula for dot product of two vectors
a · b = |a| |b| cos(θ)
formula for the angle between two vectors using dot product
θ = arccos((A · B) / (||A|| ||B||))
