Final Review Flashcards
sin30 or sin(π/6)
1/2
sin45 or sin(π/4)
rt2/2
sin60 or sin(π/3)
rt3/2
cos30 or cos(π/6)
rt3/2
cos45 or cos(π/4)
rt2/2
cos60 or cos(π/3)
1/2
tan30 or tan(π/6)
rt3/3
tan45 or tan(π/4)
1
tan60 or tan(π/3)
rt3
csc30 or csc(π/6)
2
csc45 or csc(π/4)
rt2
csc60 or csc(π/3)
(2rt3)/3
sec30 or sec(π/6)
(2rt3)/3
sec45 or sec(π/4)
rt2
sec60 or sec(π/3)
2
cot30 or cot(π/6)
rt3
cot45 or cot(π/4)
1
cot60 or cot(π/3)
rt3/3
reference angle
distance between angle’s terminal side and the x-axis
coterminal angle
two angles of different measures that share a terminal side
Pythagorean theorem
a^2 + b^2 = c^2
sine
opposite / hypotenuse
cosine
adjacent / hypotenuse
tangent
opposite / adjacent
cosecant
inverse of sin; hypotenuse / opposite
secant
inverse of cosine; hypotenuse / adjacent
cotangent
inverse of tangent; adjacent / opposite
sine equation
y = asin(b(x - c)) + d
starts on midline and goes up
a (trig equations)
amplitude (absolute value)
distance from max/min points to midline
b (trig equations)
frequency –> affects period
period = 2π/b
c (trig equations)
phase / horizontal shift
d (trig equations)
vertical shift (midline)
period of sine function
2π or 360
plot points by dividing period by 4
sine function domain / range
domain: (-inf, +inf)
range: midline +/- amplitude
cosine equation
y = acos(b(x - c)) + d
begins at max point (begins at minimum point if reflected)
period of cosine function
2π or 360
cosine function domain / range
domain: (-inf, +inf)
range: midline +/- amplitude
tangent function
y = atan(b(x - c)) + d
period of tangent function
π or 180
to plot points, put midline point halfway between asymptotes, 30 point 1/4 away, and 60 point 3/4 away
tangent function domain / range
domain: x ≠ (π/2) +/- πn
asymptotes at 1/2 period from the start +/- period(n)
range: (-inf, +inf)
cosecant function
y = acsc(b(x - c)) + d
graph points identical to sine function, but every point that would be on midline becomes a vertical asymptote
cosecant function domain / range
domain: x ≠ 0 +/- (1/2period)n
range: (-inf, min point] u [max point, +inf)
secant function
y = asec(b(x - c)) + d
graph identical to cosine, but place asymptotes at every midline point
secant function domain / range
domain: x ≠ (π/2) +/- (1/2period)n
range: (-inf, min point] u [max point, +inf)
cotangent function
y = acot(b(x - c)) + d
behaves identical to tangent, but begins on left and goes down; is also shifted over by π/2
cotangent function domain / range
domain: x ≠ 0 +/- (period)n
range: (-inf, +inf)
how to get rid of sin / cos / tan
sin-1, cos-1, tan-1
for sec: flip value, use cos-1
for csc: flip value, use sin-1
for cot: flip value, use tan-1
cscθ
1/sinθ
secθ
1/cosθ
cotθ
1/tanθ
sinθ
1/cscθ
cosθ
1/secθ
tanθ
1/cotθ
tanθ
sinθ/cosθ
cotθ
cosθ/sinθ
sin^2θ + cos^2θ
1
sec^2θ
1 + tan^2θ
csc^2θ
1 + cot^2θ
sin(a+b)
sinacosb + sinbcosa
sin(a-b)
sinacosb - sinbcosa
cos(a+b)
cosacosb - sinasinb
cos(a+b)
cosacosb + sinasinb
tan(a+b)
(tana + tanb) / (1 - tanatanb)
tan(a-b)
(tana - tanb) / (1 + tanatanb)
sin(2a)
2sinacosa
cos(2a)
cos^2a - sin^2a
1 - 2sin^2a
2cos^2a - 1
tan(2a)
(2tana) / (1 - tan^2a)
law of sines
sinA/a = sinB/b = sinC/c
law of cosines
a^2 = b^2 + c^2 -2bccosA
b^2 = a^2 + c^2 - 2accosB
c^2 = a^2 + b^2 -2abcosC
ambiguous case
remember when using inverse trig! there might be two triangles
vector
quantity with magnitude (length) and direction (angle made with positive x-axis)
bearing
angle made with positive y-axis
component form of a vector
< x, y >
< vcosθ, vsinθ >
magnitude
rt(x^2 + y^2)
direction
tan-1(y/x)
resultant vector
a1 + a2 = b
< x, y > + < x, y > = < x, y >
rectangular form equation
a + bi
polar form equation
r(cosθ + isinθ)
product of complex numbers
z1z2 = r1r2(cos(θ1 + θ2) + isin(θ1 + θ2))
quotient of complex numbers
z1/z2 = (r1/r2)(cos(θ1 - θ2) + isin(θ1 - θ2))
De Moivre’s Theorem
z^n = r^n(cos(nθ) + isin(nθ))
equation of a circle
r^2 = (x - h)^2 + (y - k)^2
converting between polar and rectangular
(x, y) = (r, θ)
x^2 + y^2 = r^2
rcosθ = x
rsinθ = y
polar line goes through radius
θ = a
polar circle centered at origin
r = a
polar circle shifted horizontally
r = acosθ
polar circle shifted vertically
r = asinθ
rose
r = acos(nθ)
r = asin(nθ)
n is odd = n leaves
n is even = 2n leaves
angle between leaves in a rose
2pi / # of leaves
parametric equation line
x (t) = x + Δxt
y (t) = y + Δyt
parametric equation circle
x (t) = h + rcos(bt)
y (t) = k + rsin(bt)
quadratic/projectile motion
x (t) = (vcosθ)t + x
y(t) = -16t^2 + (vsinθ)t + y
circle
same radii
ellipse
different radii
((x-h)^2 / a^2) + ((y-k)^2 / b^2) = 1
((y-k)^2 / a^2) + ((x-h)^2 / b^2) = 1
ellipse foci
c^2 = a^2 - b^2
hyperbola
((x-h)^2 / a^2) - ((y-k)^2 / b^2) = 1
– opens horizontally
((y-k)^2 / a^2) - ((x-h)^2 / b^2) = 1
– opens vertically
special fact about hyperbola radii
the largest radii doesn’t have to be on the transverse axis –> it can be “-y” and y can still have a larger a/b value
hyperbola foci
c^2 = a^2 + b^2
