anonymous
  • anonymous
Write the expression as either the sine, cosine, or tangent of a single angle. cosine of pi divided by five times cosine of pi divided by seven plus sine of pi divided by five times sine of pi divided by seven.
Mathematics
  • Stacey Warren - Expert brainly.com
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
ignore that, ill post a screenshot
anonymous
  • anonymous
mathmale
  • mathmale
Where have you seen cos a cos b + sin a sin b before? What does it represent?

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jim_thompson5910
  • jim_thompson5910
Hint: look on page 2 of this PDF http://tutorial.math.lamar.edu/pdf/Trig_Cheat_Sheet.pdf look at the "Sum and Difference Formulas" section
mathmale
  • mathmale
Can you think of an equivalent expression for cos a cos b + sin a sin b?
mathmale
  • mathmale
Think: trig identities!
anonymous
  • anonymous
sin (a+b)?
mathmale
  • mathmale
Hello, @plorb! You're on the way, but have a ways to go yet. Double check the accuracy of your answer.
anonymous
  • anonymous
oh is it cos (a+b)?
mathmale
  • mathmale
Double check that also. Also look at cos (a-b). Which one is it?
mathmale
  • mathmale
cos (a+b) = ? cos (a-b) + ?
anonymous
  • anonymous
cos a+b = cos a + cos b + sin a + sin b
mathmale
  • mathmale
Afraid not. cos (a+b)=cos a cos b - sin a sin b. cos (a-b) + ?
anonymous
  • anonymous
im so lost ,sorry
mathmale
  • mathmale
sorry. cos (a-b) = ?
jim_thompson5910
  • jim_thompson5910
@plohrb do you see the "Sum and Difference Formulas" section of that PDF I posted?
anonymous
  • anonymous
yeah, im following the sum and difference formula but mathmale said it wasnt correct so im confused :(
mathmale
  • mathmale
I'm very sorry. Look up on the 'Net: Sum and difference formulas for sine and cosine The point I've been trying to convey is that cos (a-b) = cos a cos b + sin a sin b.
mathmale
  • mathmale
It's important that you make a list of this and other trig identities for frequent reference.
anonymous
  • anonymous
i see
jim_thompson5910
  • jim_thompson5910
ok hopefully you see this on that PDF \[\Large \cos(\alpha \pm \beta) = \cos(\alpha)\cos(\beta) \mp \sin(\alpha)\sin(\beta)\]
anonymous
  • anonymous
yes thats the one ive been looking at
mathmale
  • mathmale
Now apply that formula, that is, the formula for cos (a-b), to the given expression (you poosted that as an image).
mathmale
  • mathmale
posted, not poosted. ;)
jim_thompson5910
  • jim_thompson5910
the symbol \(\Large \pm\) means "plus or minus". Notice how the "plus" is on top the symbol \(\Large \mp\) means "minus or plus". The "minus" is now on top the reason why these two symbols are used is that if you have a plus on the left side, then you'll have a minus on the right side. Or vice versa
anonymous
  • anonymous
im still very lost, what does the formula apply to the problem ?
jim_thompson5910
  • jim_thompson5910
see attached
1 Attachment
anonymous
  • anonymous
okay?
jim_thompson5910
  • jim_thompson5910
do you see how we have a pattern of "cos * cos ... sin * sin" ?
anonymous
  • anonymous
yes i see that
jim_thompson5910
  • jim_thompson5910
so that's why we're using that identity: it matches up
jim_thompson5910
  • jim_thompson5910
there is a plus on the right side, so you have to use a minus on the left side
anonymous
  • anonymous
yes
anonymous
  • anonymous
now we substitute? or what
jim_thompson5910
  • jim_thompson5910
what are \(\Large \alpha\) and \(\Large \beta\) going to be?
anonymous
  • anonymous
5 and 7?
jim_thompson5910
  • jim_thompson5910
btw \(\Large \alpha\) = greek letter alpha \(\Large \beta\) = greek letter beta
jim_thompson5910
  • jim_thompson5910
not 5 and 7, but you're on the right track in a way
anonymous
  • anonymous
pi/5 and pi/7?
jim_thompson5910
  • jim_thompson5910
yes
jim_thompson5910
  • jim_thompson5910
you replace all of the argument
anonymous
  • anonymous
yay
mathmale
  • mathmale
@plohrb : Why not write the given equation, and then, below it, write the identity that Jim has been discussing. Then it should be easier to identify the values of his alpha and beta.
mathmale
  • mathmale
I'm so glad you have already identified your alpha and beta correctly.
anonymous
  • anonymous
i just stated the alpha and beta right?
mathmale
  • mathmale
Yes, right.
mathmale
  • mathmale
So, now you want to write cos (alpha - beta), substituting the values you've identified.
anonymous
  • anonymous
cos (pi/5 - pi/7)
mathmale
  • mathmale
If alpha is Pi/5 and beta is Pi/7, what is the difference? subtract the latter from the former.
mathmale
  • mathmale
Yes. You could either leave your answer like that or use the LCD to combine the two angles into one.
anonymous
  • anonymous
wait so thats the simpe answer? or do i have to the sin part as well
mathmale
  • mathmale
Try this: \[combined.\angle=\pi(1/5 - 1/7)\]
jim_thompson5910
  • jim_thompson5910
yep you got it @plohrb \[\Large \cos\left(\alpha - \beta\right) = \cos\left(\alpha\right)\cos\left(\beta\right) + \sin\left(\alpha\right)\sin\left(\beta\right)\] \[\Large \cos\left({\color{red}{\alpha}} - {\color{blue}{\beta}}\right) = \cos\left({\color{red}{\alpha}}\right)\cos\left({\color{blue}{\beta}}\right) + \sin\left({\color{red}{\alpha}}\right)\sin\left({\color{blue}{\beta}}\right)\] \[\Large \cos\left({\color{red}{\frac{\pi}{5}}} - {\color{blue}{\frac{\pi}{7}}}\right) = \cos\left({\color{red}{\frac{\pi}{5}}}\right)\cos\left({\color{blue}{\frac{\pi}{7}}}\right) + \sin\left({\color{red}{\frac{\pi}{5}}}\right)\sin\left({\color{blue}{\frac{\pi}{7}}}\right)\] now you just simplify the expression \(\Large \frac{\pi}{5}-\frac{\pi}{7}\)
mathmale
  • mathmale
No, you don't need the sine function here.
anonymous
  • anonymous
ah, so the answer is just the pi/5 - pi/7
mathmale
  • mathmale
Technically, yes. But you are asked to express your result in terms of JUST ONE angle. Thus, please combine Pi/5 - Pi/7. Alternatively, evaluate this?
mathmale
  • mathmale
\[\pi(\frac{ 1 }{ 5 }-\frac{ 1 }{ 7})\]
mathmale
  • mathmale
Hint: the LCD here is 5*7, or 35.
anonymous
  • anonymous
oh um pi/35?
jim_thompson5910
  • jim_thompson5910
\(\Large \frac{\pi}{5}-\frac{\pi}{7}\) is NOT equal to \(\Large \frac{\pi}{35}\). Try again
mathmale
  • mathmale
Please show all your steps. Your numerator is incorrect.
anonymous
  • anonymous
2pi / 35?
mathmale
  • mathmale
Rewrite both Pi/5 and Pi/7 so that they have the LCD, which is 35. Note that you must also modify the numerators. Great job!!!1
mathmale
  • mathmale
So, what is your final answer? Please go back and look at the question before responding.
anonymous
  • anonymous
cos = 2pi/35.
mathmale
  • mathmale
"Write the following in terms of the sine, or cosine, or tangent, of a single angle. Take out that " = " symbol and write what's left. 2pi/35 is the "argument" of the cosine function.
mathmale
  • mathmale
The given expression, involving the sine and cosine, can be expressed as the ________ (name of trig function) of a single angle, and that angle is ___________.
anonymous
  • anonymous
its just cos 2pi/35 then?
mathmale
  • mathmale
Perfect. Congrats!
jim_thompson5910
  • jim_thompson5910
I'd use parenthesis and say `cos(2pi/35)`
jim_thompson5910
  • jim_thompson5910
but yes you have the answer
mathmale
  • mathmale
You will see these trig identies again and again.
mathmale
  • mathmale
Worth writing them down and reviewing them regularly so that you know them immediately when you need them.
anonymous
  • anonymous
thanks to both of you! have a great evening :)
mathmale
  • mathmale
My great pleasure. And thanks so much to you, Jim, for your thoughtful input.

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