math 2: trigonometric functions Flashcards
P(x, y) is any point on the
terminal side of the angel
r is the distance between
O and P
sinx =
y/r
cosx=
x/r
tanx=
y/x
cotx=
x/y
secx=
r/x
cscx=
r/y
sinx · cscx =
1
cosx · secx =
1
tanx · cotx =
1
tanx = sinx/
cosx
cotx= cosx/
sinx
cos²θ + sin²θ =
1
sin²θ =
1- cos²θ
cos²θ =
1-sin²θ
sec²θ =
1 + tan²θ
csc²θ =
cot²θ + 1
All Students
Take Calculus
angles of Q. II, IIII, or IV have
reference angles θr
any trig function of θ =
± the same function of θr (sign determined by quadrant)
sin & cos, tan & cot, sec & csc are
cofunction pairs
confuctions of complementary angles are
equal
if a and b are complementary, then
triga=cotrig(b) & trigb=cotrig(a)
one radian is about
57.3 degrees
length of an arc, s, is:
rθ
area of sector AOB is
1/2r²θ
domain of sine:
(-∞, +∞)
range of sine:
-1≤ sinx ≤1
domain of cosine:
(-∞, +∞)
range of cosine:
-1≤ cosx ≤1
domain of tan:
{x| x≠ π/2 + πn, where n is an integer}
range of tangent:
all real numbers
domain of cot:
{x| x≠ πn, where n is an integer}`
range of cotangent:
all real numbers
domain of secant:
{x| x≠ π/2 + πn, where n is an integer}
range of secant:
(-∞, -1] ∪ [1, +∞)
domain of csc:
{x| x≠ πn, where n is an integer}
range of csc:
(-∞, -1] ∪ [1, +∞)
general form of a trig function: y=
A · trig(Bx + C) + D
amplitude→
|A|
horizontal translation (phase shift) →
-C/B
horizontal dilation (period)
either 2π/B or π/B
vertical translation→
D
(reciprocal identities) cscx =
1/sinx
(reciprocal identities) secx=
1/cosx
(reciprocal identities) cotx=
1/tanx
(cofunction identities) sinx=
cos(π/2 - x)
(cofunction identities) secx=
csc(π/2 - x)
(cofunction identities) tanx=
cot(π/2 - x)
(cofunction identities) cosx=
sin(π/2 - x)
(cofunction identities) cscx=
sec(π/2 - x)
(cofunction identities) cotx=
tan(π/2 - x)
(double angle identities) sin2x=
2(sinx)(cosx)
(double angle identities) cos2x (both sin and cos)
= cos²x - sin²x
(double angle identities) cos2x (just cos)
= 2cos²x - 1
(double angle identities) cos2x (just sin)
= 1 - 2sin²x
graphs of the inverses of trig functions aren’t graphs of functions, so the domain neeeds to be limited to one period for
range values to be achieved exactly once
sin^-1 domain
[-1, 1]
sin^-1 range
[-π/2, π/2]
cos^-1 domain
[-1, 1]
cos^-1 range
[0, π]
tan^-1 domain
(-∞, +∞)
tan^-1 range
(-π/2, π/2)
cot^-1 domain
(-∞, +∞)
cot^-1 range
(0, π)
sec^-1 domain
(-∞, -1] ∪ [1, +∞)
sec^-1 range
[0, π/2) ∪ (π/2, π]
csc^-1 domain
(-∞, -1] ∪ [1, +∞)
csc^-1 range
[-π/2, 0) ∪ (0, π/2]
