Question

Please help. WILL MEDAL! @surjithayer @mathmale @ganeshie8 @Hero @TheSmartOne @sleepyjess @Preetha @SolomonZelman @zepdrix @rebeccaxhawaii @Vocaloid @dtan5457 @MrNood

Answer 1

\[-2 + \sqrt{3} \sec \theta = 0\]

Answer 2

We communicated about this problem earlier. What happened? Why repost it?

I've suggested that you replace that "sec theta" temporarily with the variable z. How would you then solve for z? Please show your work.

Answer 3

I'm clueless on how to do this type of problem.

Answer 4

Can someone teach me how to do this?

Answer 5

@mathmale You didn't actually help me solve it at all.

Answer 6

As mathmale said

\[-2 + \sqrt{3}*z = 0\]

You wanted to find what sec(theta) is, so instead find what z is first by using algebra

Answer 7

Please solve this equation for z.

Answer 8

I don't understand.

Answer 9

All right. Please add 2 to both sides of this equation. Your result?

Answer 10

It's the same as finding z for something like:

\[1+z=2\]

Answer 11

\[\sqrt{3} \times z = 2\]

Answer 12

Good. Now, divide both sides by sqrt(3) and you will then have solved for z.

Answer 13

\[z = \frac{ 2 }{ sqrt{3} } \]

Answer 14

Very good. Now, replace that 'z' with sec theta.

Answer 15

z \[z = \frac{ 2\sqrt{3} }{ 3 }\]

Answer 16

\[\sec \theta = \frac{ 2\sqrt{3} }{ 3 }\]

Answer 17

Yes. Since you are probably more comfortable with the cosine function than you are with the secant function, try inverting both sides of the equation.

sec theta becomes 1 / sec theta = cos theta.

Can you finish this work?

Answer 18

Left side: cos theta.
Right side = ?

Answer 19

The instructions tell me to solve the following equations given that \[0degrees \le \theta < 360 degrees\]

Answer 20

Yes. We're getting there. what's the right side going to look like?

Answer 21

\[\frac{ 2\sqrt{3} }{ 3 }\]
Why this? I think I missed something

Answer 22

I'm confused again.

Answer 23

I got lost here: secθ=23√3

Answer 24

Let me quickly summarize what we've done so far:
\[-2 + \sqrt{3} \sec \theta = 0\]

Answer 25

can be solved for sec theta: sec theta = 2/sqrt(3).

Answer 26

\[\sec \theta = \frac{ 2 }{ \sqrt{3} }\]

Answer 27

Agree or not?

Answer 28

I understand

Answer 29

Good. Now, invert both sides. Recall that sec theta = 1 / cos theta.

Answer 30

\[\sec \theta = \frac{ 2\sqrt{3} }{ 3 }\]

Answer 31

This is where I'm stuck.

Answer 32

I don't know what to do next.

Answer 33

"Invert" means "to turn something upside down."

Inverting "sec theta" produces 1 over sec theta = 1 / sec theta

and inverting 2 / sqrt(3) produces sqrt(3) / 2. Clear on this?

Answer 34

I understand that cosine is the inverse function of secant

Answer 35

Sorry to say "no" on that, but that's not the case; you meant " cosine is the reciprocal of the secant."

Answer 36

oh yes good catch

Answer 37

:3

Answer 38

You understand what I mean.

Answer 39

Inverting "sec theta" produces 1 over sec theta = 1 / sec theta

and inverting 2 / sqrt(3) produces sqrt(3) / 2. Clear on this?

then we g et 1 / sec theta = cos theta = sqrt(3) / 2:

Answer 40

\[\cos \theta = \frac{ \sqrt{3} }{ 2 }\]

Answer 41

So, \[ \frac{ 1 }{ \sec \theta } = \frac{ \sqrt{3} }{ 2 }\]

Answer 42

When I see this latest equation, I think immediately: since cos theta = adjacent side / hyp,

cos theta in this problem = sqrt(3) / 2, and adj side =sqrt(3), and hyp = 2. Clear, or need more discussion?

Answer 43

Okay I understand.

Answer 44

clear

Answer 45

\[\frac{ 1 }{ \sec \theta } = \frac{ \sqrt{3} }{ 2 }=\cos \theta\]

Answer 46

gottit

Answer 47

Yes. You merely replace 1/sec theta with cos theta.

Answer 48

understood

Answer 49

so now we have:\[\cos \theta = \frac{ \sqrt(3) }{ 2 }\]

Answer 50

Okay, I'm still with you.

Answer 51

we can interpret this to mean that cos theta = adj. side / hypotenuse = sqrt(3) / 2.
Thus, I claim that the hyp has a length of 3 and the adj side has a length of sqrt(3). OK?

Answer 52

Sorry, hyp has length 2.

Answer 53

Understood.

Answer 54

OK. While not strictly necessary, sketching this angle might help you figure out what the angle theta is.

Answer 55

So it is a 30-60-90 triangle.

Answer 56

Yes, that's correct. adj side is sqrt(3); hyp is 2. Which angle have we here (theta)?

30? 60? 90?

Answer 57

So now I have to find all of the values between 0 and 360 degrees

Answer 58

30

Answer 59

You're saying that the cosine of 30 degrees is sqrt(3) / 2. True. If I weren't here to say, "True," how would you check that cos 30 actually equals sqart(3) / 2?

Answer 60

How can I tell if there are any other values?

Answer 61

I'm not sure how to check.

Answer 62

Put 30 into the cosine function?

Answer 63

Experimentation.

30 degrees , as measured from the positive x-axis, has an adj side of sqart(3) and a hyp of 2. What would happen if you were to graph -30 degrees? what would the adj side be in that case? what would the length of the hyp be in that case?

Answer 64

The sides would be the same length at -30 degrees.

Answer 65

If you were to put 30 deg into the cosine function, you'd just get sqrt(3)/2, or approx. 0.866.