Question

θ is any real number. is it possible for sinθ=.25 and cosθ=.5? why or why not?

Answer 1

so is this the same for this problem?
-1 ≤ sin θ ≤ 1

Answer 2

^yes. I'm assuming that they're not implying the θ in sinθ is the same θ as in cosθ.

Answer 3

oh ok thanks

Answer 4

-1 ≤ cos θ ≤ 1
this too.

Answer 5

hm i'm not sure what they mean

Answer 6

I think the "θ is any real number" just means θ can be anything necessary to satisfy the sinθ=.25 and cosθ=.5, doesn't have to be the same value (i'm guessing).

Answer 7

you're probably right. so in that case how can i find the answer?

Answer 8

If they do mean that it has to be the same value, you'd have to take the inverse sine for sinθ=.25 and inverse cosine for cosθ=.5

Answer 9

what would that be?

Answer 10

@agent0smith is there a formula i should use to find it?

Answer 11

Have you used inverse sin (written as sin^-1) and inverse cosine (cos^-1) yet? Maybe arcsin and arccos? If neither of those seem remotely familiar, then I wouldn't worry about them for now.

Answer 12

yes i think i have i just dont really remember it too clearly

Answer 13

im just not sure what the answer should be here

Answer 14

sinθ=.25
inverse sin of both sides gives
θ = arcsin0.25 (do this on your calculator)

cosθ = 0.5
inverse cosine of both sides gives
θ = arccos 0.5

Answer 15

I would argue that the language "is it possible for sinθ=.25 AND cosθ=.5? Why or why not?" means they want you to determine whether or not there is a value of theta for which both are true.

Answer 16

^ yeah that's what i started to think too. The "and" hints at that.

Answer 17

i got θ=.2527

Answer 18

And a quick glance at the neighborhood unit circle says that for that to be true, .25^2 + .5^2 must = 1^2.

Answer 19

oh ok i see what you mean

Answer 20

It's not a very interesting question otherwise...

Answer 21

so i should say that there is or is not a value of theta which is true for both?

Answer 22

Is the equation that I said must be true, true?

Answer 23

^correct.

"i got θ=.2527" - this would be in radians, not degrees.

Answer 24

remember, this is just that basic trig identity sin^x + cos^x = 1

Answer 25

@whpalmer4 Idk if they'd have done trig identities yet (but maybe). The last question she posted was basic, just "is it possible for sinθ=5/4".

Answer 26

ahh im so confused :/ sorry :(

Answer 27

@agent0smith well, that's not inconsistent whether or not they've had the identities so long as they've had the unit circle.

Answer 28

Erin, have you seen the unit circle showing you the relationship between sin and cos?

Answer 29

yes

Answer 30

Ah, then it probably is asking you if there's a θ that satisfies both cos = 0.25 and sin = 0.5.

Answer 31

And you could use what whpalmer4 posted above... sin^2 θ + cos^2 θ =1

Answer 32

Okay, in that case, I think the question is as I suggest: they want to know if there is some angle for which both of those statements are true at the same time.

And using the Pythagorean theorem in the unit circle (hypotenuse = 1, cos x = a, and sin x = b), you can see that for there to be such an angle, 0.25^2 + 0.5^2 = 1^2. If that is not true, then there is no such angle.

Answer 33

What is (1/2)^2 + (1/4)^2?

Answer 34

.3125

Answer 35

so its not true

Answer 36

ohhh i understand now. thank you both so much, i really appreciate your help

Answer 37

Good!

Answer 38

can you please help me with another onee? :)

Answer 39

If you are working in radians, for small values of \(\theta\), \(\sin \theta \approx \theta\). You saw that when you found 0.2527 for \(\theta\) but perhaps you didn't realize it :-)

Answer 40

oh i understand now, thanks so much