Question

Please help! Reduce cos(30degrees-theta) - cos(30degrees+theta) to a single term.
a. sintheta
b. sqrt3 sintheta
c. -sintheta
d. -sqrt3 sintheta

Answer 1

use the formula for cos(A+B) and the one for cos(A-B)

Answer 2

do you know these formulas?

Answer 3

Can you please give them to me? :/

Answer 4

have a look on this site: http://www.mathsisfun.com/algebra/trigonometric-identities.html

scroll down to the section titled: Angle Sum and Difference Identities

Answer 5

its near the very bottom of the page

Answer 6

ughh I tried solving it but I failed:

answer is sin theta

http://www.wolframalpha.com/input/?i=%28cos+30+cos+theta+%2B+sin+30+sin+theta%29+-+%28cos30+cos+theta+-+sin30+sin+theta%29

If you could explain how to solve it, I will be grateful asnaseer.

Answer 7

@JoshDavoll you should not give the answer out. It is better to guide the user to the answer. That way they learn more.

Answer 8

Well I've been trying to solve it as well thanks to your guidance. Can you please explain though?

Answer 9

Oh sorry, asnaseer.

Answer 10

what have you tried so far @IloveCharlie ?

Answer 11

@JoshDavoll so you used wolfram to solve?

Answer 12

if you list each step then I can try and spot where you may have made a mistake.

Answer 13

@IloveCharlie - yeah, I broke the problem into cos(x+y) = cosxcosy-sinxsiny, and cos(x-y) = cosxcosy+sinxsiny.

Answer 14

Yes, same. I used the formula and link asnaseer provided.

Answer 15

but after breaking them down to those identities, I got stuck :)

Answer 16

you are both thinking along the right lines.

so what you need to do is to convert this:

cos(30degrees-theta) - cos(30degrees+theta)

into:

cos(x-y) - cos(x+y)

and expand this using the formulas

Answer 17

here I have let x=30, y=theta

Answer 18

@IloveCharlie can you try doing that and let me know what you get?

Answer 19

Just a sec

Answer 20

ok :)

Answer 21

When I plugged it into my calc I got 0. I plugged it into wolfram as well http://www.wolframalpha.com/input/?i=cos%2830-theta%29+-+cos%2830%2Btheta%29

Answer 22

no - do NOT use a calculator - you don't need to use one here - just use algebra

Answer 23

we know cos(x+y) = cos(x)cos(y) - sin(x)sin(y)
and cos(x-y) = cos(x)cos(y) + sin(x)sin(y)

use these expansions to work out what cos(x-y) - cos(x+y) = ?

Answer 24

and leave the result in terms of x and y.

Answer 25

(cos30cos(theta)+sin30sin(theta)) - (cos30cos(theta)-sin30sin(theta))
right?

Answer 26

work with x and y please

Answer 27

oh right

Answer 28

we can substitue in the correct values at the very end

Answer 29

@IloveCharlie do you understand or do you want me to explain in more detail?

Answer 30

(cosxcosy+sinxsiny) - (cosxcosy - sinxsiny)
right?

Answer 31

yes - and please let ILoveCharlie work this out.

Answer 32

oh okay..sure..

Answer 33

Oh yes! I understand now! Thanks so much for the help and explanation!

Answer 34

so what do you get for the expression cos(x-y) - cos(x+y) = ?

Answer 35

sintheta! :)

Answer 36

:/ I wasn't asking for the final answer, I wanted to know what you got for:

cos(x-y) - cos(x+y) = ?

Answer 37

Well I was able to plug in x and y to the formula and got sintheta. http://www.wolframalpha.com/input/?i=%28cos30costheta%2Bsin30sintheta%29+-+%28cos30costheta+-+sin30sintheta%29

Answer 38

remember wolfram will not be with you in the exams

Answer 39

:/ sorry. I shouldn't have said anything before..

Answer 40

np @JoshDavoll - at least you understand why now :)

Answer 41

asnaseer, can you still walk me through it? :)

Answer 42

sure...

Answer 43

so, we started with:

we know cos(x+y) = cos(x)cos(y) - sin(x)sin(y)
and cos(x-y) = cos(x)cos(y) + sin(x)sin(y)

use these expansions to work out what cos(x-y) - cos(x+y) = ?

Answer 44

thanks! so reduces to (cosxcosy+sinxsiny) - (cosxcosy - sinxsiny)

Answer 45

this gives us:

cos(x-y) - cos(x+y) = [cos(x)cos(y) + sin(x)sin(y)] - [cos(x)cos(y) - sin(x)sin(y)]
= cos(x)cos(y) + sin(x)sin(y) - cos(x)cos(y) + sin(x)sin(y)
= 2sin(x)sin(y)

agreed?

Answer 46

sorry, how did the operator change in the 2nd line to a '+' ?

Answer 47

because:

-( a - b ) = -a -(-b) = -a + b

Answer 48

two minuses make a plus

Answer 49

ahh cool, ok I understand, and how did you get to 2sinxsiny?

Answer 50

e.g.:

-( 5 - 2 ) = -5 + 2 = -3

Answer 51

^because the - distributes right?

Answer 52

cos(x-y) - cos(x+y) = [cos(x)cos(y) + sin(x)sin(y)] - [cos(x)cos(y) - sin(x)sin(y)]
= cos(x)cos(y) + sin(x)sin(y) - cos(x)cos(y) + sin(x)sin(y)
= cos(x)cos(y) - cos(x)cos(y) + sin(x)sin(y) + sin(x)sin(y)
= 2sin(x)sin(y)

Answer 53

Ahh! right, I understand all of that.

Answer 54

^commutative law right?

Answer 55

good, so finally if we now plug in the actual values we get:\[\cos(30-\theta)-\cos(30+\theta)=2\sin(30)\sin(\theta)\]

Answer 56

(yes commutative law)

Answer 57

and we know sin(30) = 0.5

Answer 58

therefore:\[\cos(30-\theta)-\cos(30+\theta)=2\sin(30)\sin(\theta)=2\times 0.5\times\sin(\theta)=\sin(\theta)\]

Answer 59

:) Awesome explaining. Thank you very much, asnaseer.

Answer 60

np - and I hope you were also following along IloveCharlie :)

Answer 61

it may seem hard at first trying to learn it this way but the benefits will be enormous when it comes to exam time :)

Answer 62

Yes, I have been. Thank you so much both of you!

Answer 63

yw :)