Find number of solns to 2 sin x = cos x in (0, 90)

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rainyday2003

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0* < x < 90* , how many solutions are there for the equation 2 sin x =cos x?

I know the answer is 2 (from my study guide) but I can not find out how it is two.
I need someone to explain it in stupid terms for me. I need to know what they are asking and how to find the answer. It is not homework , I am studing for my ASSET test in a week. Any help would be welcomed . Thank you for your time.

(the * are actually degrees,you know the little round circle in the upper right corner of the angle??)
 
I do not know this is exactly how it appears on the study paper;
3. For 0*<x< 90*, how many solutions are there for the equation 2 sin x = cos x ?

just a reminder * is for the degree sign.

I have spent hours amoung hours trying to figure it out.

Thank you so much for helping.
 
Oh, OK. That's different. You had a typo in the first post.

\(\displaystyle \L\\2sin(x)=cos(x)\)

\(\displaystyle \L\\2tan(x)=1\)

\(\displaystyle \L\\tan(x)=\frac{1}{2}\)

\(\displaystyle \L\\x=n{\pi}+tan^{-1}(\frac{1}{2})\)

I may be mistaken, but it appears there is only one solution in your given interval and that is when n=0.
 
Your right it does only have one solution but I do not know how to solve it or where I can go to learn to solve it.I really want to explain how it is solved. I am so confused.And my goal is that I have to figure out how to solve it.
You did solve it but how did you do it?
Thank you for trying to help me.
 
If you understand the previous answer then all you need is the solution to \(\displaystyle x =\arctan ( \frac {1}{2})\). Because the arctangent function is one-to-one the solution is unique.
 
I know the answer is 2 (from my study guide) but I can not find out how it is two.
Click to expand...

However, the answer is not 2.

- i.e.

(number of solutions) is not equal to 2
 
rainyday2003 said:
I do not know this is exactly how it appears on the study paper;
3. For 0*<x< 90*, how many solutions are there for the equation 2 sin x = cos x ?

just a reminder * is for the degree sign.

I have spent hours amoung hours trying to figure it out.

Thank you so much for helping.
Click to expand...

Ok....

2 sin x = cos x

sin x = (1/2) cos x

sin x / cos x 1/2

tan x = 1/2

AND, you're told that 0 <= x <= 90

You're asked HOW MANY solutions this equation has. For an angle between 0 and 90 degrees, there's JUST ONE angle x for which tan x = 1/2.
 
If you don't like the tangent.

\(\displaystyle \L\;2\sin(x) + \cos(x) = \sqrt{5}\cos(x-atan(2))\)

Of course, you still haven't told us the original equation, so it's a bit difficult to proceed.
 
the questions is:(* is for the degree sign)

For 0* < x < 90*, how many solutions are there for the equation 2 sin x = cos x ?
I know the answer is one but how is it solved?How is it figured out?

I have no idea what they are asking . I do go and look in the book and online for sin and cos but can not understand what they are saying and their examples do not look like mine.I have not be in school for 7 years i do not understand it. i need a book for dummy's so it would be easier for someone like me to understand this .
I just relized with my other problem I had made it way harder than it was.
Thank you so much for all your time.
 
Two step-by-step solutions were posted. I am sorry, but what else can we do?. Good luck.

You have \(\displaystyle \L\\2sin(x)=cos(x)\)

Divide through by cos(x) and this gives:

\(\displaystyle \L\\\frac{2sin(x)}{cos(x)}=\frac{cos(x)}{cos(x)}\)

But \(\displaystyle \L\\\frac{sin(x)}{cos(x)}=tan(x)\)

Therefore, you have:

\(\displaystyle \L\\2tan(x)=1\)

Divide through by 2:

\(\displaystyle \L\\tan(x)=\frac{1}{2}\)

Inverse of tan is arctan and you have:

\(\displaystyle \L\\x=tan^{-1}(\frac{1}{2})\approx{0.463647609001}\)

If you want to convert to degrees, multiply by \(\displaystyle \L\\\frac{180}{\pi}\)

This gives you \(\displaystyle \L\\26.5650511771=26^{\circ}33'54.18"\) degrees.

That's the only solution between 0 and 90 degrees.

Check this back in the original equation:

2sin(26.5650511771)=0.894427191

cos(26.5650511771)=0.894427191

There ya' go. They're the same, so that must be a solution. Okey-doke?.
 
rainyday2003 said:
the questions is:(* is for the degree sign)

For 0* < x < 90*, how many solutions are there for the equation 2 sin x = cos x ?
Click to expand...
I see. If I were paying attention, I may have said...

\(\displaystyle \L\;2\sin(x) - \cos(x) = \sqrt{5}\sin(x-atan(1/2))\)

I wondered why I was getting a 2 instead of a 1/2, like everyone else.
 
I think the question is more basic than the steps. He's either asking about the logic of the approach (as in, what motivates us to divide through by cos, or subtract the cos) or about what sin and cos refer to...
 
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