trigonometry Flashcards by Arina Zhdanova

coterminal angles

angles with the same initial and terminal side
infinitely many

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angle in standard position

the initial side lies on the x-axis
vertex of the angle is in the origin (0,0)

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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

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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

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what is a radian?

value of an angle in which the angle is equal to the corresponding arc length on the unit circle

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calculating the length of the arc subtending the angle

s = rad x r
s … length of the arc
rad … angle in radians
r … radius

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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

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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

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tan, cot, sec, cosec

tan = sin/cos
cot = cos/sin
sec = 1/cos
cosec = 1/sin

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what is a periodic function?

f(x) = f(x + w)
w … the basic period (smallest positive period)

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sin 0°, 0 rad

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sin 30°

1/2

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sin π/6

1/2

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sin π/4

√2/2

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sin 45°

√2/2

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sin 60°

√3/2

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sin π/3

√3/2

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sin 90°

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sin π/2

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cos 0°, 0

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cos 30°

√3/2

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cos π/6

√3/2

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cos π/4

√2/2

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cos 45°

√2/2

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cos 60°

1/2

cos π/3

1/2

cos π/2

cos 90°

tan 0

tan 30°

√3/3

tan π/6

√3/3

tan 45°

tan π/4

tan 60°

√3

tan π/3

√3

tan 90°

∅

tan π/2

∅

cot 0

∅

cot 30°

√3

cot π/6

√3

cot 45°

cot π/4

cot 60°

√3/3

cot π/3

√3/3

cot 90°

cot π/2

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