trigonometry Flashcards
coterminal angles
angles with the same initial and terminal side
infinitely many
angle in standard position
the initial side lies on the x-axis
vertex of the angle is in the origin (0,0)
a vs the unit circle
a unit circle => any circle with a radius of 1
the unit circle => a unit circle with its center in the origin
what is a degree?
1/360 of the complete angle
Babylonian system was base 60
1° is the angle of the Earth’s path according to the Sun each day
what is a radian?
value of an angle in which the angle is equal to the corresponding arc length on the unit circle
calculating the length of the arc subtending the angle
s = rad x r
s … length of the arc
rad … angle in radians
r … radius
area of a sector
A = area of a circle x rad/2pi = rad x r^2 / 2 = rs/2
area of a circle … pi r^2
rad … angle in radians
r … radius
s … length of the arc
sin and cos def
sin is the y-coordinate of the intersection of an angle with the unit circle
cos is the x-coordinate of this intersection
tan, cot, sec, cosec
tan = sin/cos
cot = cos/sin
sec = 1/cos
cosec = 1/sin
what is a periodic function?
f(x) = f(x + w)
w … the basic period (smallest positive period)
sin 0°, 0 rad
0
sin 30°
1/2
sin π/6
1/2
sin π/4
√2/2
sin 45°
√2/2
sin 60°
√3/2
sin π/3
√3/2
sin 90°
1
sin π/2
1
cos 0°, 0
1
cos 30°
√3/2
cos π/6
√3/2
cos π/4
√2/2
cos 45°
√2/2
cos 60°
1/2
cos π/3
1/2
cos π/2
0
cos 90°
0
tan 0
0
tan 30°
√3/3
tan π/6
√3/3
tan 45°
1
tan π/4
1
tan 60°
√3
tan π/3
√3
tan 90°
∅
tan π/2
∅
cot 0
∅
cot 30°
√3
cot π/6
√3
cot 45°
1
cot π/4
1
cot 60°
√3/3
cot π/3
√3/3
cot 90°
0
cot π/2
0
complementary angle identities
sin(π/2 - α) = cosα
cos(π/2 - α) = sinα
tan(π/2 - α) = cotα
cot(π/2 - α) = tanα
supplementary angle identities
sin(π - α) = sinα
cos (π - α) = -cosα
tan (π - α) = -tanα
cot(π - α) = -cot α
formulas of the third quadrant
sin(π + α) = - sinα
cos (π + α) = - cosα
tan (π + α) = tanα
cot(π + α) = cot α
opposite angle identities
sin(-α) = - sinα
cos(-α) = cosα
tan(-α) = -tanα
cot(-α) = -cotα
f(x) = sinx
domain, range, period, extrema, roots, even/odd
D: all real numbers
R: [-1,1]
w: 2π
minima: ( -π/2 + 2kπ, -1)
maxima: (π/2 + 2kπ, 1)
roots: x = kπ
odd function
f(x) = cosx
domain, range, period, extrema, roots, even/odd
D: R
R: [-1, 1]
w: 2π
minima: (π + 2kπ, -1)
maxima: (2kπ, 1)
roots: x = π/2 + kπ
even function
sinosoidal function
stretched and translated sin => f(x) = Asin(B(x - C)) + D
A…amplitude
B…frequency
C…phase shift
D…vertical shift
how do we find the amplitude, frequency, vertical shift of a sinosoidal function?
A = (max-min)/2
B = w(initial)/w(translated) => sinusoidal function: B = 2π/w
D = (max+min)/2
order of transformations in sinusoidal functions
sinx => Asinx => AsinBx => AsinB(x-C) => AsinB(x-C) + D
f(x) = tanx
range, roots, poles, period
R: R
roots: x = kπ
poles: x = π/2 + kπ
w: π
solutions of cosx = c
c < -1, c > 1 => x ∈ ∅
c = 1 => x = 2kπ
c = -1 => x = π + 2kπ
c = 0 => x = π/2 + kπ
1 < c < 0 => x1= arccosc + 2kπ; x2 = -arccosc + 2kπ
solutions of sinx = c
c < -1, c > 1 => x ∈ ∅
c = 1 => x = π/2 + 2kπ
c = -1 => x = -π/2 + 2kπ
c = 0 => x = kπ
1 < c < 0 => x1 = arcsinc + 2kπ; x2 = π - arcsinc + 2kπ
solutions of tanx = c
x = arctanx + kπ
1 + tan2α; 1 + cot2α
1 + tan2α = 1/cos2α
1 + cot2α = 1/sin2α
cos(α±β) =
cosα cosβ ∓ sinα sinβ
sin(α±β) =
sinα cosβ ± cosα sinβ
tan(α±β) =
(tanα±tanβ)/(1∓ (tanα)(tanβ))
sin2x =
2sinx cosx
cos 2x =
(cosx)^2 - (sinx)^2
2(cosx)^2 - 1
1 - 2(sinx)^2
tan2x =
2tanx/(1 - (tanx)^2)
sin(x/2)
= ± √((1-cosx)/2)
cos(x/2) =
= ± √((1+cosx)/2)
sinx + siny =
= 2 sin(x+y/2) cos(x-y/2)
cosx + cosy =
2 cos(x+y/2) cos(x-y/2)
f(x) = arcsinx
domain, range, odd/even, key points, increasing/decreasing
D: [-1, 1]
R: [-π/2, π/2]
odd function
key points: (0, 0), (1, π/2), (-1, -π/2)
increasing
f(x) = arccosx
domain, range, odd/even, key points, increasing/decreasing
D: [-1, 1]
R: [0, π]
neither odd nor even
key points: (1, 0), (0, π/2), (-1, π)
decreasing
arcsinx + arccosx =
= π/2
f(x) = arctanx
domain, range, key points, even/odd, increasing/decreasing
D: R
R: [-π/2, π/2]
always increasing
(0,0), (1, π/4), (-1, -π/4)
sin(arccosx) =
√(1 - x^2)
cos(arcsinx) =
√(1-x^2)
tan(arccosx) =
(√(1-x2))/x
arccos(sinx) =
π/2 - x
arcsinx(cosx) =
π/2 - x
