Test 2

Question 1

Original graph for y = sin(x)

Answer 1

x | 0 | π/2 | π | 3π/2 | 2π |

y | 0 | 1 | 0 | -1 | 0 |

Question 2

Original graph for y = cos(x)

Answer 2

x | 0 | π/2 | π | 3π/2 | 2π |

y | 1 | 0 | -1 | 0 | 1 |

Question 3

Equations for y = cos(x) & y = sin(x)

Answer 3

A(trig F)(ωx - φ) + V

Question 4

How to find period

Answer 4

2π / ω = P for sin & cos; π / ω = P for tan

Question 5

How do you know whether to use cos/sin from a graph on the test?

Answer 5

cos starts above/below 0;

sin goes through origin

Question 6

Equation for amplitude from graph

Answer 6

(max - min) / 2

Question 7

Equation for period from graph

Answer 7

2π/(cycle end - cycle start)

Question 8

Original graph for y = tan(x)

Answer 8

x | -π/2 | -π/4 | 0 | π/4 | π/2 |

y | NA | -1 | 0 | 1 | NA |

Question 9

Graph for tan (where are asymptotes & period too)

Answer 9

|      ||
                     |     ||
                     |   //
------------------//-------------------
               //    |
             ||       |
            ||        |
asymptotes @ ±xπ/2; period = π

Question 10

Graph for cot (where are the asymptotes, period, & x-intercept too)

Answer 10

|  ||     
                     |   || 
                     |    \\
-------------------------\\------------
                     |          \\
                     |            ||
                     |             ||
asymptotes @ ±xπ & 0; period = π; x-int = π/2

Question 11

How to draw sec graph & its asymptotes

Answer 11

1. Draw cos graph
2. At each crest & troph, reciprocate it into a bunch of parabolas sticking up & down
3. Points on cos graph w/ a y-val of 0 will be vertical asymptotes (so ±xπ/2)

Question 12

How to draw csc graph & its asymptotes

Answer 12

1. Draw sin graph
2. At each crest and troph, reciprocate it into a bunch of parabolas sticking up & down
3. Points on sin graph w/ a y-val of 0 will be vertical asymptotes (so ±xπ & 0)

Question 13

Order of transformations

Answer 13

period, phase shift, amplitude, vertical shift

Question 14

What happens if ω is negative?

Answer 14

Undef for sin & tan; if it’s in cos, use even function property and…take the negative away lmao

Question 15

How do you know whether Amp is positive or negative?

Answer 15

If the cycle begins or goes straight to a negative troph, it’s a negative Amp
If it begins/goes straight to positive troph, it’s positive

Question 16

Original domain & range of sin⁻¹(x)

Answer 16

D = [-π/2, π/2]
R = [-1, 1]

Question 17

f⁻¹(f(x)) –> sin⁻¹(sin(x)) –> what within what boundaries

Answer 17

x radians

-π/2 ≤ x ≤ π/2

Question 18

f(f⁻¹(x)) –> sin(sin⁻¹(x)) –> what within what boundaries

Answer 18

x ratio

-1 ≤ x ≤ 1

Question 19

What if angle in an inverse trig F breaks the pi limits?

Answer 19

take the ref angle instead but MAKE SURE THE SIGN OF THE REF ANGLE IS EQUAL TO THE ONE THATS PLUGGED IN OK??

Question 20

What if ratio in inverse trig F breaks [-1, 1] limits?

Answer 20

it’s undefined

Question 21

Original domain & range of cos⁻¹(x)

Answer 21

D = [-1, 1]
R = [0, π]

Question 22

f⁻¹(f(x)) –> cos⁻¹(cos(x)) –> what within what boundaries

Answer 22

x radians

0 ≤ x ≤ π

Question 23

f(f⁻¹(x)) –> cos(cos⁻¹(x)) –> what within what boundaries

Answer 23

x ratio

-1 ≤ x ≤ 1

Question 24

Original domain & range of tan⁻¹(x), and vertical asymptotes

Answer 24

D = (-∞, ∞)
R = [-π/2, π/2]
VA = ±π/2

Question 25

f⁻¹(f(x)) –> tan⁻¹(tan(x)) –> what within what boundaries

Answer 25

x radians

-π/2 ≤ x ≤ π/2

Question 26

f(f⁻¹(x)) –> tan(tan⁻¹(x)) –> what within what boundaries

Answer 26

x ratio

-∞ ≤ x ≤ ∞ (so all the time)

Question 27

should you designate if an inverse of a function = f⁻¹(x) on a test?

Answer 27

YES FREAKING DO IT

Question 28

Original domain & range of sec⁻¹(x), and vertical asymptotes

Answer 28

D = (-∞, -1)⋃(1, ∞)
R = [0, π]
VA = xπ/2

Question 29

Original domain & range of csc⁻¹(x), and vertical asymptotes

Answer 29

D = (-∞, -1)⋃(1, ∞)
R = [-π/2, π/2]
VA = 0

Question 30

Original domain & range of cot⁻¹(x), and vertical asymptotes

Answer 30

D = ∞
R = [0, π]
No VA

Question 31

Range group for [0, π]

Answer 31

cos
sec
cot

Question 32

Range group for [-π/2, π/2]

Answer 32

sin
csc
tan

Question 33

How to find inverse of csc & sec

Answer 33

Just flip the ratio inside the parentheses & use the corresponding range group sin or cos function

Question 34

Steps to take if you get a negative ratio for cot

Answer 34

1. Take the ratio, flip it, and put it in tan⁻¹
2. Plug into calc and get an angle
3. If in degrees, subtract from 180; if in radians, subtract from xπ

Question 35

How to write something like sin( tan⁻¹(u)) in an expression in terms of u

Answer 35

1. Take the ratio u/1 as tangent and put it as the lengths of triangle sides, then work from there