3b Flashcards
inverse of a function
the result of switching the input and output values of a function
the output of an inverse trigonometric function will be…
an angle measure
two ways to represent inverse trigonometric functions
1) sin^-1
2) arcsin
why are domains of inverse trigonometric functions restricted?
they wouldn’t be functions without the restrictions! (wouldn’t pass the vertical line test! (normal trigonometric functions don’t pass horizontal line test))
to have inverses, a function must pass…
the horizontal line test
restricted domain of an sin function
[-pi/2, pi/2]
restricted domain of an cos function
[0, pi]
restricted domain of an tan function
(-pi/2, pi/2)
the restricted domain of a function is the ____ of its inverse function
range of possible solutions
restricted domains as shaded on unit circle
sin ~ Q1, Q4 (right half)
cos ~ Q1, Q2 (top half)
tan~ Q1, Q4 (right half)
after find solutions of an inverse function…
double check they are in the allotted domain!
is sin(x)=1/2 equivalent to arcsin(1/2)=x
No, both have pi/6 as a solution but due to the restricted domain of the inverse function that is its only solution whereas the sin function has multiple (infinite) solutions.
since trigonometric functions are periodic…
they can have infinitely many solutions (if the domain is not restricted)
general solution to sin(x)=1/2
x=pi/6+2pik
and
x=5pi/6+2pik
where k is any integer
general solution to cos(x)=1/2
x=pi/3+2pik
and
x=5/3+2pik
where k is any integer
general solution to tan(x)=1
x=pi/4+pik
where k is any integer
general solution to all sin and cos functions
-find solutions for x on first rotation of unit circle
-write an equation for each with +2pik attached
-clarify k is any integer
general solution to all tan functions
-find smallest solution for x on first rotation of unit circle
-write an equation with +pik attached to the solution
-clarify k is any integer
steps to solve trig equations
- isolate the trig function
- find the corresponding angles that satisfy the equation
- consider the domain restrictions
–if note add pik or 2pik - write solutions
REMEMBER WITH SQUARE ROOTS
solutions can be +/-
(sinx)^n=
sin^nx
[] vs ()
[] if </> or equal to (includes end values)
() if strictly </> (does not include max and min values)
steps for solving trig inequalities
- set f(x)=0 and solve
- make a sign chart with you solutions (remember domain restrictions!)
- test a value in each interval
- interpret the sign chart and answer the inequality
</> signs flip when…
you divide by a negative number
secant
sec=1/cos
cosecant
csc=1/sin
cotangent
cot=1/tan=cos/sin
Graphs of sec, csc, cot are undefined where…
sec, csc, tan hit x axis
range of sec
(-inf, -1] [1, inf)
graph of sec and csc
mini parabolas on top of maxs and mins
undefined ares of sec graph
x=pi/2+pik
range of csc
(-inf, -1] [1, inf)
range of cot
(-inf, inf)
undefined areas of csc graph
x=pik
undefined areas of cot graph
x=pik
cot graph
negative version of tan graph shifted over pi/2, so it fits the new asymptotes of pik
to find asymptotes of an equation
plug x and the transformations that directly affect it into the “x” of the equation and solve so x is by itself
to solve csc, cot, sec
isolate and change into sin, cos, tan to use unit citcle
pythagorean identity
cos^2x+sin^2x=1
you can change the pythagorean identity
into a lot!
–even tan, if you divide by cos!
–to solve for sin^2x or cos^2x
sum and diff identities
sin(a+/-b)=sinacosb+/-cosasinb
- sign is the same
cos(a+/-b)=cosacosb-/+sinasinb
- sign changes
double angle identities
sin(2x)=2sinxcosx
cos(2x)=cos^2x-sin^2x=1-2sin^2x=2cos^x-1
polar coordinates
r, theta
output, input
y =
sin(theta) * r
x =
cos(theta) *r
r =
+/- sqrt (x^2 + y^2)
tan(theta) =
y/x
i =
sqrt(-1)
complex number rectangular coordinates
a+bi = (a,b)
complex number polar coordinates
(rcos(theta))+i(rsin(theta))
r(theta) = asin(theta), +/- a?
+ ~ symmetrical to y axis on top
- ~ symmetrical to y axis on bottom
r(theta) = acos(theta), +/- a?
+ ~ symmetrical to x axis on right
- ~ symmetrical to x axis on left
if b is even…
graph will have 2 b petals
if b is odd…
graph will have b petals
if r is positive and increasing, the distance between r and the origin is…
increasing
if r is negative and decreasing, the distance between r and the origin is…
increasing
if r is positive and decreasing, the distance between r and the origin is…
decreasing
if r is negative and increasing, the distance between r and the origin is…
decreasing
relative extrema occur
when graph changes from increasing to decreasing or vice versa
AROC of a polar function
y2-y1/x2-x1
indicates the rate at which the radius is changing per radian
estimating with AROC
f(x) = known y + AROC (x-known x)
