Question

Find the values of x for which the equation cos x = -1 is true. k represents an integer.

Answer 1

there is no k in your question

Answer 2

7

Answer 3

I got the last one

Answer 4

cos(x)=-1
cos(x)+1=0

That means that your x-solutions are the roots of
y=cos(x)+1
https://www.desmos.com/calculator/ahvced9xlt

Answer 5

What do you mean by roots

Answer 6

roots of the functions are the zeros (or the x-intercepts) of the function.

Answer 7

I still don't get it honestly. Do I have to find the zeros of the equation

Answer 8

@jim_thompson5910

Answer 9

for each answer choice, let's replace k with some whole number

k = 0 is probably the easiest to deal with

Answer 10

so instead of 2pi*k, it is 2pi*k = 2pi*0 = 0

what happens when you plug x = 0 into the original equation?

Answer 11

well, cos(0) = 1

Answer 12

is this basically plugging in?

Answer 13

so `cos(x) = -1` turns into `1 = -1` when you replaced x with 0 and evaluated the left hand side

Answer 14

`1 = -1` is a false equation, so x = 0 is NOT a solution (it is not in the solution set)

Answer 15

I plugged in pi + 2pi(k) and got -1

Answer 16

did you first replace k with 0 and then simplify to get pi?

Answer 17

No. Oops. So I have to do cos(*insert equation*) and then have k be 0?

Answer 18

do you understand why I'm replacing k with 0?

Answer 19

bc it represents an integer?

Answer 20

yes, k is any integer and 0 is the easiest to work with because 0 makes things go away

pi+2pi*k = pi+2pi*0 = pi + 0 = pi

so `pi+2pi*k` is equivalent to `pi` when k = 0

Answer 21

you'll then find `cos(pi) = -1`

so...

cos(x) = -1
cos(pi) = -1 ... replace x with pi
-1 = -1 ... true equation

because the last equation is true, the first equation is true when x = pi

so x = pi is definitely one of the infinitely many solutions

x = pi is in the solution set for the equation cos(x) = -1

Answer 22

So how do I figure out which one is right?

Answer 23

do you agree that x = pi is a solution?

Answer 24

Well, that's one of the solutions I got as well so yes

Answer 25

ok try k = 1 and plug it into pi+2pi*k

what is `pi+2pi*k` equivalent to when k = 1 ?

Answer 26

9.4 :/

Answer 27

in terms of pi

Answer 28

so it would be like 2*1 or 3.14 + 2*1?

Answer 29

pi+2pi*k = pi + 2pi*1 = ???

treat pi as if it were a variable

Answer 30

2pi?

Answer 31

think of it like

2 pies + 1 pie = how many pies?

Answer 32

3 pies lol

Answer 33

yep so 2pi+pi = 3pi

Answer 34

if k = 1, then pi+2pi*k = 3pi

Answer 35

tell me what cos(x) is equal to when x = 3pi

Answer 36

-1

Answer 37

now try k = 2. Plug it into pi+2pi*k to get what?

Answer 38

So I would just add the pi's the same way?

Answer 39

yeah it's similar to pi+2pi

Answer 40

so would it be 4pi?

Answer 41

pi+2pi*k = pi+2pi*2 = ?????

Answer 42

6? cause we're keeping it in terms of pi?

Answer 43

ill brb I have to workout

Answer 44

pi+2pi*k = pi+2pi*2 = pi+4pi = ?????

Answer 45

5pi?

Answer 46

cos(5pi) = -1 too

Answer 47

yep, let's do one more

if k = 3, then pi+2pi*k is equal to what?

Answer 48

8pi?

Answer 49

just replace k with 3 and simplify

\[\Large \pi+2\pi*\color{red}{k}=\pi+2\pi*\color{red}{3} = ???\]

Answer 50

wait, no 9

Answer 51

try again

Answer 52

wouldn't it be 9pi though? I don't see how it qwouldnt be