Question

How do I solve arctan[cos(7pi/2)]? Cos of 7pi/2 isn't on the unit circle.

Answer 1

you know
7pi/2 =3pi/2
right

Answer 2

What? 6 pi/2 = 3/2... But not 7 pi... Right..? I don't understand.

Answer 3

you know that cos(x) = cos(x + k(2pi)) right?

Answer 4

what do you think cos(7pi/2) equals to in terms of cos( *number here* ) ?

Answer 5

Should I just add 2pi? 21 pi / 2..?

Answer 6

well if you add, though the value of cosine stays the same, the angle is bigger. Do you think that would help? what is the range of angle we usually use the most?

Answer 7

0-2pi

Answer 8

is 7pi/2 in that range?

Answer 9

I still don't understand where to plug in 21 pi / 2 though

Answer 10

Yeah it is

Answer 11

Oh wait, I need to subtract 2 pi?

Answer 12

BINGO!

Answer 13

ohhhh. 3pi/2

Answer 14

That makes a lot more sense. Thank you

Answer 15

so cos(7pi/2) = cos(7pi/2 - 2pi) yes?

Answer 16

Yeah. So that means after I find the cosine value and substitute it for the inverse tangent, the answer radian is undefined?

Answer 17

nope. What did you get for cos(3pi/2) ?

Answer 18

Oh, 0.

Answer 19

excellent. What is arctan(0) ?

Answer 20

0+pi k?

Answer 21

well no, 0 in this case is no longer the angle, why?

Answer 22

it's the radian measure?

Answer 23

Is it just 0?

Answer 24

correct it's just 0 (a number). For example, sin(pi/6) = 1/2. pi/6 is the angle. What is 1/2 then? well, just a number for sine when the angle is pi/6. Make sense?

Answer 25

Yeah, so I wouldn't have to call it a radian or an angle measure... But isn't Pi another number that answers arctan (0)? 0/-1 = 0...

Answer 26

you're right. As soon as we evaluate arctan(0), which equals 0, <-- this 0 is actually now the angle in radian.

just like, arctan(1/sqrt(2)) = pi/4, <-- this is the angle.
arctan(0) = 0 < this is the angle
..........^
this is not.

Answer 27

OK that makes sense. But what if I was asked to find the angles where tan = a value, like 3.7321?

Answer 28

Is that the same sort of problem, solving for angles?

Answer 29

Unfortunately, finding the angle by hand is subject to limitation. There are special cases like cos(Θ) = 1/2, tan(Θ) = 1/sqrt(2) ... where we can easily find the angles using the unit circle. However, given any random number k, e.g.
cos(Θ) = k, tan(Θ) = k, etc...it's likely that we'd have to rely on technology, like a calculator.

Answer 30

Okay. What about cos x = -0.8?

Answer 31

you'd need a calculator for that one

Answer 32

Alright. How would I know when to add +2pi k or +pi k? For a question like cos x = sqrt(3)/2, I could find that pi/6 is the radian value, but would I add 2pi k to that problem, or just leave it at pi/6?

Answer 33

well, cos(x), sin(x), tan(x) are period, so you'll need to add the period. In the case of cos(x) = sqrt(3)/2
x = pi/6 + k(2pi). There are also x values but I don't say it here

Answer 34

...are *periodic*

Answer 35

you know how cos(x) = cos(x + k(2pi)) ? then if x=t is the root, the ALL roots will be x = t + k(2pi)

Answer 36

Okay. I get that. How can I solve for x if I've got Cos^2 (x) = 1/2?

Answer 37

how would you get rid of the square?

Answer 38

Take the square root. So Sqrt 2 / 2?

Answer 39

close. What happens when you take a square root?

Answer 40

+ or -?

Answer 41

very good. So what do you have?

Answer 42

+ or - sqrt 2 / 2....+ 2 pi k?

Answer 43

no, sqrt(2)/2 is just the number. You only add k(2pi) to the roots.

Answer 44

oh okay. What about a more complicated equation, like
sec^2 (x) + 3 tan^2 (x) = 13
I can already tell that I want to use the Pythagorean identity that Tan^2 (x) + 1 = Sec^2 (x), but I can't tell where to start the problem. Dividing out the 3 would just create problems with the Secant, wouldn't it?

Answer 45

The truth is that you can't possibility solve every single trig question out there. There are many messy trig equations that, when trig identities are used, amazingly reduced to the ones we can work with. But that is still limited. So the case of the function above, you well need to use a graphing calculator.

Answer 46

I just realized we can solve the equation above by hand O.O. But then still, say you have something like cos^3(x) - sin(5x) cot^3(x) - 1 = 0, then a graphing calculator is guaranteed.

Answer 47

I see. But on this problem, where would I start? Dividing the 3 out would be wrong, would it not?

Answer 48

this is an example where a trig identity is used to reduced a messy equation to a nice equation.
sec^2(x) = tan^2(x) + 1

sec^2 (x) + 3 tan^2 (x) = 13
[tan^2(x) + 1] + 3tan^2(x) = 13
4tan^2(x) + 1 = 13
4tan^2(x) = 12
tan^2(x) = 3

as you can see, that's pretty nice.

Answer 49

Now I see it, is there any thing else I can do afterwards to solve for x in the final equation? Tan^2(x)=3?

Answer 50

take the square root

Answer 51

tan x = sqrt (3)
What now?

Answer 52

tan(x) = sqrt(3) or tan(x) = -sqrt(3)

Answer 53

can I find x?

Answer 54

tan(x) = sqrt(3)
what angle would that be?

Answer 55

pi/3?

Answer 56

there is one more within [0,2pi]

Answer 57

4pi/3?

Answer 58

good. What is the period of tan(x) ?

Answer 59

pi k
so, +pi k

Answer 60

One more question, I need to simplify this equation. I'm pretty sure it simplifies by hand.
Sin^2 (x) + sin x - 2 = 0

Answer 61

correct,
x = pi/3 + k(pi) or
x = 4pi/3 + k(pi)

you exactly the same in the case of tan(x) = -sqrt(3)

Answer 62

Alright

Answer 63

can you factor u^2 + u - 2 ?

Answer 64

OH it's factoring
I didn't even catch that

Answer 65

yes, this is why it's really also important to know how factor when you encounter a problem like this

Answer 66

I would have to use Quadratic formula, right?

Answer 67

You could, though it can be factored as (u - 1) (u+2)

Answer 68

are you sure? Isn't there a coeffecient of two in front of u^2 though? I don't follow...

Answer 69

I believe you're were hallucinating :P