Question

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.

Answer 1

ignore that, ill post a screenshot

Answer 2

Where have you seen cos a cos b + sin a sin b before? What does it represent?

Answer 3

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

Answer 4

Can you think of an equivalent expression for cos a cos b + sin a sin b?

Answer 5

Think: trig identities!

Answer 6

sin (a+b)?

Answer 7

Hello, @plorb! You're on the way, but have a ways to go yet. Double check the accuracy of your answer.

Answer 8

oh is it cos (a+b)?

Answer 9

Double check that also. Also look at cos (a-b).

Which one is it?

Answer 10

cos (a+b) = ?

cos (a-b) + ?

Answer 11

cos a+b = cos a + cos b + sin a + sin b

Answer 12

Afraid not. cos (a+b)=cos a cos b - sin a sin b.

cos (a-b) + ?

Answer 13

im so lost ,sorry

Answer 14

sorry. cos (a-b) = ?

Answer 15

@plohrb do you see the "Sum and Difference Formulas" section of that PDF I posted?

Answer 16

yeah, im following the sum and difference formula but mathmale said it wasnt correct so im confused :(

Answer 17

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.

Answer 18

It's important that you make a list of this and other trig identities for frequent reference.

Answer 19

i see

Answer 20

ok hopefully you see this on that PDF

\[\Large \cos(\alpha \pm \beta) = \cos(\alpha)\cos(\beta) \mp \sin(\alpha)\sin(\beta)\]

Answer 21

yes thats the one ive been looking at

Answer 22

Now apply that formula, that is, the formula for cos (a-b), to the given expression (you poosted that as an image).

Answer 23

posted, not poosted. ;)

Answer 24

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

Answer 25

im still very lost, what does the formula apply to the problem ?

Answer 26

see attached

Answer 27

okay?

Answer 28

do you see how we have a pattern of "cos * cos ... sin * sin" ?

Answer 29

yes i see that

Answer 30

so that's why we're using that identity: it matches up

Answer 31

there is a plus on the right side, so you have to use a minus on the left side

Answer 32

yes

Answer 33

now we substitute? or what

Answer 34

what are \(\Large \alpha\) and \(\Large \beta\) going to be?

Answer 35

5 and 7?

Answer 36

btw
\(\Large \alpha\) = greek letter alpha
\(\Large \beta\) = greek letter beta

Answer 37

not 5 and 7, but you're on the right track in a way

Answer 38

pi/5 and pi/7?

Answer 39

yes

Answer 40

you replace all of the argument

Answer 41

yay

Answer 42

@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.

Answer 43

I'm so glad you have already identified your alpha and beta correctly.

Answer 44

i just stated the alpha and beta right?

Answer 45

Yes, right.

Answer 46

So, now you want to write cos (alpha - beta), substituting the values you've identified.

Answer 47

cos (pi/5 - pi/7)

Answer 48

If alpha is Pi/5 and beta is Pi/7, what is the difference? subtract the latter from the former.

Answer 49

Yes. You could either leave your answer like that or use the LCD to combine the two angles into one.

Answer 50

wait so thats the simpe answer? or do i have to the sin part as well

Answer 51

Try this: \[combined.\angle=\pi(1/5 - 1/7)\]

Answer 52

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 \(\Large \frac{\pi}{5}-\frac{\pi}{7}\)

Answer 53

No, you don't need the sine function here.

Answer 54

ah, so the answer is just the pi/5 - pi/7

Answer 55

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?

Answer 56

\[\pi(\frac{ 1 }{ 5 }-\frac{ 1 }{ 7})\]

Answer 57

Hint: the LCD here is 5*7, or 35.

Answer 58

oh um pi/35?

Answer 59

\(\Large \frac{\pi}{5}-\frac{\pi}{7}\) is NOT equal to \(\Large \frac{\pi}{35}\). Try again

Answer 60

Please show all your steps. Your numerator is incorrect.

Answer 61

2pi / 35?

Answer 62

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

Answer 63

So, what is your final answer? Please go back and look at the question before responding.

Answer 64

cos = 2pi/35.

Answer 65

"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.

Answer 66

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 ___________.

Answer 67

its just cos 2pi/35 then?

Answer 68

Perfect. Congrats!

Answer 69

I'd use parenthesis and say `cos(2pi/35)`

Answer 70

but yes you have the answer

Answer 71

You will see these trig identies again and again.

Answer 72

Worth writing them down and reviewing them regularly so that you know them immediately when you need them.

Answer 73

thanks to both of you! have a great evening :)

Answer 74

My great pleasure. And thanks so much to you, Jim, for your thoughtful input.