Trigonometric Identities Flashcards
which function does sin have a reciprocal relationship with?
csc
which function does cos have a reciprocal relationship with?
sec
which function does tan have a reciprocal relationship with?
cot
which function does csc have a reciprocal relationship with?
sin
which function does sec have a reciprocal relationship with?
cos
which function does cot have a reciprocal relationship with?
tan
power reduction of sin^2(u)
1/2(1-cos(2u))
power reduction of cos^2(u)
1/2(1+cos(2u))
power reduction of tan^2(u)
(1-cos(2u)) / (1+cos(2u))
double angle identity: sin(2u)
2sinucosu
double angle identity: cos(2u)
cos^2(u) - sin^2(u)
pythagorean identity: tan^2(u)
sec^2(u) - 1
pythagorean identity: sec^2(u)
tan^2(u) + 1
substitution: sqrt(a^2 + u^2)
u = (a)(tan(theta))
substitution: sqrt(a^2 - u^2)
u = (a)(sin(theta))
substitution: sqrt(u^2 - a^2)
u = (a)(sec(theta))
arc length s of curve f(x)
s = definite integral from a to b of sqrt(1 + (f’(x))^2) dx
Trap (n) = , also delta x =
(1/2)(delta x)[f(x,0) + 2f(x,1) + 2f(x,2) + … … + 2f(x,n-1) + f(x,n)]
delta x = (upper bound of integral - lower bound of integral)/(number of sub intervals ie terms in series)
Mid (n) = , also delta x =
(delta x)[f(x,1) + f(x,2) … … + f(x,n)]
delta x = (upper bound of integral - lower bound of integral)/(number of sub intervals ie terms in series)
Simp (n) = , also delta x =
(1/3)(delta x)[f(x,0) + 4f(x,1) + 2f(x,2) + 4f(x,3) + 2f(x,4) … … 4f(x,n-1) + f(x,n)]
n must be an even number.
separable differential equation
dy/dx = (f(y))(g(x))
when derivative is isolated the other side of the equation can be factored so that one factor is a function of only y and the other factor is a function of only x.
define implicit form of a differential equation
implicit form is not solved for y in terms of x (y is not completely isolated on one side)
define explicit form of a differential equation
explicit form is solved for y in terms of x (y is completely isolated on one side)
Integral Identity: integral of u^n du , when n does not equal 1
u^(n+1) / n+1 + C
Integral Identity: integral of u^-1 du
ln|u| + C
Integral Identity: integral of e^u
e^u + C
Integral Identity: integral of a^u , when a does not equal 1
(1 / (lna) ) (a^u) + C
Trig Integral Identity: integral of cos(u)
sin(u) + C
Trig Integral Identity: integral of sin(u)
-cos(u) + C
Trig Integral Identity: integral of (sec u)(tan u) du
“a sea can tan if a sea can can.”
sec(u) + C
Trig Integral Identity: integral of sec^2(u) du
“A sea can square its t….”
tan(u) + C
“A sea can square its toes”
Trig Integral Identity: integral of (csc u)(cot u) du
“a cosy cot can if a cosy can can’t.”
-csc(u) + C
Trig Integral Identity: integral of csc^2(u) du
“a cozy can square without a cot there.”
-cot(u) + C
Trig Integral Identity: integral of tan(u) du
“that tan has no lines cuz!”
-ln| cos(u) | + C
Trig Integral Identity: integral of cot(u) du
“that cot lines its signs!”
ln| sin(u) | + C
Trig Integral Identity: integral of sec(u) du
“A sea can always lines it’s sea cans with tans.”
ln| sec(u) + tan(u) | + C
Trig Integral Identity: integral of csc(u) du
“A cosy can alone can’t line it’s coats with cosy cans.”
-ln| csc(u) + cot(u) | + C
Trig Integral Identity: integral of 1 / sqrt(a^2 - u^2) du
sin^-1 (u/a) + C
Trig Integral Identity: integral of 1 / sqrt(a^2 + u^2) du
(1/a)(tan^-1 (u/a)) + C
Integral Identity: integral of ln(u)
(x)(ln(x)) - x + C
Integral Identity: integral of (u)(dv) , ie two statements multiplied
(u)(v) - integral of (v)(du)
Trig Integral Identity: integral of 1 / ((u)(sqrt(u^2 - a^2) ) )
(1/a)(sec^-1( | u/a | ) + C
Derivative of sin^-1 (u)
u’ / sqrt(1 - u^2 )
Derivative of tan^-1 (u)
u’ / (1 + u^2 )
Derivative of sec^-1 (u)
(u’) / ( |u| )(sqrt(u^2 - 1) )
Pythagorean Identity of cot^2(theta)
csc^2(theta) - 1
Pythagorean Identity of csc^2(theta)
cot^2(theta) + 1
circle identity of sin(theta)
y/r
circle identity of cos(theta)
x/r
circle identity of tan(theta)
y/x
circle identity of csc(theta)
r/y
circle identity of sec(theta)
r/x
circle identity of cot(theta)
x/y
cofunction identity (ie pi/2 - theta) of sin(theta)
cos(pi/2 - theta)
cofunction identity (ie pi/2 - theta) of cos(theta)
sin(pi/2-theta)
cofunction identity (ie pi/2 - theta) of tan(theta)
cot(pi/2-theta)
cofunction identity (ie pi/2 - theta) of cot(theta)
tan(pi/2-theta)
cofunction identity (ie pi/2 - theta) of sec(theta)
csc(pi/2-theta)
cofunction identity (ie pi/2 - theta) of csc(theta)
sec(pi/2-theta)
what does sin(theta) equal in the unit circle
y
what does cos(theta) equal in the unit circle
x
what does tan(theta) equal in the unit circle
y/x
even odd property of sin(-theta)
-sin(theta)
even odd property of -sin(theta)
sin(-theta)
even odd property of cos(-theta)
cos(theta)
even odd property of cos(theta)
cos(-theta)
even odd property of -cos(theta)
-cos(theta) ie unchanged
even odd property of tan(-theta)
-tan(theta)
even odd property of -tan(theta)
tan(-theta)
even odd property of csc(-theta)
-csc(theta)
even odd property of sec(-theta)
sec(theta)
even odd property of sec(theta)
sec(-theta)
even odd property of cot(-theta)
-cot(theta)
even odd property of -cot(theta)
cot(-theta)
how do you find sin/cos/tan etc of (pi/x)?
draw a triangle with the angles (45x45x90, 30x60x90), label with unit circle values, solve pyth theorem, use soh cah toa.
sin’(x) =
x’cos(x) + C
cos’(x) =
-x’sin(x) + C
tan’(x) =
x’sec^2(x) + C
sec’(x) =
x’sec(x)tan(x) + C
csc’(x) =
-x’csc(x)cot(x) + C
cot’(x) =
-x’csc^2(x) + C
if f(x) is position what is f’(x)
velocity
if f(x) is position what is f’‘(x)
acceleration
if f(x) is position what is | f’(x) |
speed
arcsin’(x) =
1/sqrt(1-x^2) + C
arccos’(x) =
-1/sqrt(1-x^2) + C
arctan’(x) =
1/(1+x^2) + C
arccsc’(x) =
-1/(( |x| )(sqrt(x^2 - 1)) + C
arcsec’(x) =
1/(( |x| )(sqrt(x^2 - 1)) + C
arccot’(x) =
-1/(1 + x^2)
