5c pre calc

Question 1

for
a. sin
b. cos
c.tan
d. csc
e. sec
f. cot
list the domain range zeroes asymp period amp

Answer 1

a. all real numbers, -1 less then or equal to y less then or equal to 1, x = pik, none, 2pi, 1
b. all real numbers, -1 less then or equal to y less then or equal to , x = pi/2 + pik, none, 2 pi, 1
c. x cannot equal pi/2 + pik, all real numbers, x = pik, x = pi/2 + pik, pi, none
d. x cannot = pik, |y| greater then or equal to 1, none, x = pik, 2pi, none
e. x cannot equal pi/2 + pik, |y| greater then or equal to 1, none, x = pi/2 + pik, 2pi, none
f. x cannot = pik, all real numbers, x = pi/2 + pik, x = pik, pi, none

Question 2

for
a. sin ^ -1
b. cos ^ -1
c. tan ^ -1
what is the domain and range

Answer 2

a. -1 less then or equal to y less then or equal to 1,
-pi/2 less than or equal to y less than or equal to pi/2
b. -1 less then or equal to y less then or equal to 1, 0 less then or equal to y less than or equal to pi
c. all real numbers, -pi/2 less then y less then pi/2

Question 3

what is the vertical shift and phase shifts for this problem:
a. 2sin3(x-4) + 5
b. 2cos3(x+5)-4
c. what is the midline

Answer 3

a. vertical shift: up 5, phase shift: 4 right
b. vertical shift: up 5 phase shift: 5 left
c. a midline is a shifted horizontal axis -> so the new midline for a would be y = 5 so you would move the horizontal axis up 5

Question 4

how do you find the amplitude and period for this problem:
-2sin3(x-4) + 5

Answer 4

amp: |a| so its 2
period: 2pi/|b| so its 2pi/3
those equations only work for sin, cos. csc. and sec for tan and cot amp is none and period is pi/|b|

Question 5

how do you graph csc and whats its period

Answer 5

period: 2pi
asymptotes at origin, -2pi, -pi, pi, 2pi
on the right side the first section is up one in the middle and upwards u,
2nd section is down one in the middle downwards u
to the left the first section is down one downwards u, 2nd section in the middle is up one upwards u

Question 6

how do you graph sec and whats its period

Answer 6

period: 2pi
asymptotes in the middle of the origin and pi, in the middle of pi and 2pi, in the middle of origin and -pi, in the middle of -pi and -2pi
up one on the x=0 line is a upwards u,
one section to the the right is a downwards u in the middle,
next section over in the middle is a upwards u,
to the left is a downwards u in the middle and
one more to the left is a upwards u in the middle

Question 7

how do you graph cot and whats its period

Answer 7

period: pi
asymptotes at the origin pi 2pi -pi and -2pi
in the middle of those 4 sections is a dot on the line to the left the arrow goes up and to the right the arrow goes down

Question 8

graph arcsin x

Answer 8

three coordinates: (-1, -pi/2), (0, 0), (1, pi/2)
to the left line is upwards curve and after crossing the origin is curves down

Question 9

graph arccos x

Answer 9

three coordinates: (-1, pi) (0, pi/2), (1, 0)
to the left curve in downwards to the right curve is upwards

Question 10

graph arctan x

Answer 10

three coordinates: (-1, -pi/4) (0, 0) (1, pi/4)
Asymptotes at (0, pi/2) and (0, -pi/2)
left part of the line the curve is downwards right side of the line the curve is upwards

Question 11

solve
a. arccos ( square root of 3/2)
b. arcsin (-1)
c. arctan(square root of 3)

Answer 11

a. = pi/6
b. = -pi/2
c. = pi/3

Question 12

a. If _______ then sin (arcsin x) = x
b. If _______ then cos (arccos x) = x
c. if ________ then tan(arctan x) =x
d. are these always true?

Answer 12

a. -1 less then or equal to x less then or equal to 1
b. -1 less then or equal to x less then or equal to 1
c. -infinity less then or equal x less then or equal to infinity
d. yes, if the number is not in the ranges listed abouve the answer is DNE

Question 13

a. if ______ the arcsin(sin x) = x
b. if ______ then arccos(cos x) = x
c. if ______ then arctan (tan x) = x
d. are these always true?

Answer 13

a. -pi/2 less then or equal to x less then or equal to pi/2
b. 0 less then or equal to x less then or equal to pi
c. -pi/2 less then x less then pi/2
d. no, this is a nice shortcut but you can find the answer even if its not in the ranges listed above

Question 14

What is the tricks for remembering whats positive in which quadrant?

Answer 14

cast with c in the 4th quadrant

Question 15

solve arctan(tan 5pi/6)

Answer 15

since this is not in range we have to solve it the harder way
First find out what quadrant this angle would be in do is be converting it to degrees. (divide 180 by 6 then multiply that number by 5)
Then you know that the answer is negative pi/6

Question 16

solve tan(arccos squarerootof 2/2)

Answer 16

First think of which triangle these numbers correspond to the answer is the 45 45 90 triangle
so you know that the (arccos(squarerootof 2/2)) = pi/4
then you solve tan(pi/4) which means you find the tangent of 45 degrees which gives you 1